Theorem onelini 6444

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 6441	. 2 ⊢ (𝐵 ∈ 𝐴 → 𝐵 ⊆ 𝐴)
3		dfss 3922	. 2 ⊢ (𝐵 ⊆ 𝐴 ↔ 𝐵 = (𝐵 ∩ 𝐴))
4	2, 3	sylib 218	1 ⊢ (𝐵 ∈ 𝐴 → 𝐵 = (𝐵 ∩ 𝐴))

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114   ∩ cin 3902   ⊆ wss 3903  Oncon0 6325

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1089  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-v 3444  df-in 3910  df-ss 3920  df-uni 4866  df-tr 5208  df-po 5540  df-so 5541  df-fr 5585  df-we 5587  df-ord 6328  df-on 6329

This theorem is referenced by: (None)