Theorem onelini 6477

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 6474	. 2 ⊢ (𝐵 ∈ 𝐴 → 𝐵 ⊆ 𝐴)
3		dfss 3950	. 2 ⊢ (𝐵 ⊆ 𝐴 ↔ 𝐵 = (𝐵 ∩ 𝐴))
4	2, 3	sylib 218	1 ⊢ (𝐵 ∈ 𝐴 → 𝐵 = (𝐵 ∩ 𝐴))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109   ∩ cin 3930   ⊆ wss 3931  Oncon0 6357

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-ral 3053  df-v 3466  df-in 3938  df-ss 3948  df-uni 4889  df-tr 5235  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-ord 6360  df-on 6361

This theorem is referenced by: (None)