Theorem onelini 6469

Description: An element of an ordinal number equals the intersection with it. (Contributed by NM, 11-Jun-1994.)

Hypothesis

Ref	Expression
on.1	⊢ 𝐴 ∈ On

Assertion

Ref	Expression
onelini	⊢ (𝐵 ∈ 𝐴 → 𝐵 = (𝐵 ∩ 𝐴))

Proof of Theorem onelini

Step	Hyp	Ref	Expression
1		on.1	. . 3 ⊢ 𝐴 ∈ On
2	1	onelssi 6466	. 2 ⊢ (𝐵 ∈ 𝐴 → 𝐵 ⊆ 𝐴)
3		dfss 3926	. 2 ⊢ (𝐵 ⊆ 𝐴 ↔ 𝐵 = (𝐵 ∩ 𝐴))
4	2, 3	sylib 221	1 ⊢ (𝐵 ∈ 𝐴 → 𝐵 = (𝐵 ∩ 𝐴))

Syntax hints:   → wi 4   = wceq 1563   ∈ wcel 2145   ∩ cin 3906   ⊆ wss 3907  Oncon0 6350

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737

This theorem depends on definitions:  df-bi 210  df-an 401  df-3an 1103  df-tru 1566  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3080  df-v 3459  df-in 3914  df-ss 3924  df-uni 4869  df-tr 5213  df-po 5560  df-so 5561  df-fr 5605  df-we 5607  df-ord 6353  df-on 6354

This theorem is referenced by: (None)