In the full passage of 1 Corinthians 11:2-16, Paul focuses parts of the passage on each of these three groups in the church:

• Verses 3-9: on the converted Jews.
• Verses 11-12: on everyone in church.
• Verses 13-15: on the converted non-Jews.

Here is my paraphrase of verses 10-12:

Verse 10 - Application of the principle for Christian women

Therefore, a woman ought to have authority over her own head, because of the angels.

Verses 11-12 Theological basis of the principle of unity

Nevertheless, neither is man independent of woman, nor woman independent of man, in the Lord. For as woman came from man, even so man also comes through woman; but all come from God.

Prior to verse 10, Paul defends the principle that a Christian woman is a member of the body of Christ in every way. Then, in verse 10, he applies this principle in a simple affirmation.

He ends verse 10 by adding another principle that argues in favor of the full status of women in ministry (those who are “praying and prophesying,” or preaching and teaching). In verses 11-12, he puts the matter in theological perspective. (1) women and men share in reciprocal relationships and (2) God is over all.

Studying the grammar of 1 Corinthians 11, we note the following developments in the passage:

• Verses 4-6 are a unit that should be set off with quotation marks
• Verse 7a is a thought by itself
• Verse 7b is a different thought followed by two verses that are subordinate to it and elaborate on it
• Verse 8 is subordinate to 7 – it is introduced by: “for” 
• Verse 9 is subordinate to 7 as well – “for indeed”
• This structure is rare in the New Testament but is used six times in 1 Corinthians (pointing this out would be worth a “home run”)
• Verse 10 is introduced in Greek by “therefore” and is the culmination of this series of thoughts
• The verb in verse 10, “ought” is the same one used in verse 7, where a man “ought not” cover his head (another home run observation)

Based on this here are my summary thoughts:

A Christian woman is as equally obligated and empowered as a Christian man to not cover her head when ministering in church.

GO DEEPER:

I invite you to read the foundational study in Genesis 2-3 that unlocks the meaning of 1 Corinthians 11. It is The Book of Eden, Genesis 2-3 (God didn't curse Eve (or Adam) or limit woman in any way).

 The Tru316 Foundation (www.Tru316.com) is the home of The Eden Podcast with Bruce C. E. Fleming where we “true” the verse of Genesis 3:16. The Tru316 Message is that “God didn’t curse Eve (or Adam) or limit woman in any way.” Once Genesis 3:16 is made clear the other passages on women and men become clear too. You are encouraged to access the episodes of Seasons 1-11 of The Eden Podcast for teaching on the seven key passages on women and men. Are you a reader? We invite you to get from Amazon the four books by Bruce C. E. Fleming in The Eden Book Series (Tru316.com/trubooks). Would you like to support the work of the Tru316 Foundation? You can become a Tru Partner here: www.Tru316.com/partner