Theorem oneluni 6483

Description: An ordinal number equals its union with any element. (Contributed by NM, 13-Jun-1994.)

Hypothesis

Ref	Expression
on.1	⊢ 𝐴 ∈ On

Assertion

Ref	Expression
oneluni	⊢ (𝐵 ∈ 𝐴 → (𝐴 ∪ 𝐵) = 𝐴)

Proof of Theorem oneluni

Step	Hyp	Ref	Expression
1		on.1	. . 3 ⊢ 𝐴 ∈ On
2	1	onelssi 6479	. 2 ⊢ (𝐵 ∈ 𝐴 → 𝐵 ⊆ 𝐴)
3		ssequn2 4187	. 2 ⊢ (𝐵 ⊆ 𝐴 ↔ (𝐴 ∪ 𝐵) = 𝐴)
4	2, 3	sylib 221	1 ⊢ (𝐵 ∈ 𝐴 → (𝐴 ∪ 𝐵) = 𝐴)

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ∈ wcel 2144   ⊆ wss 3895   ∪ cun 4149  Oncon0 6362

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1572  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-v 3452  df-ss 3896  df-un 4151  df-uni 4867  df-tr 5212  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-ord 6365  df-on 6366

This theorem is used by:  omabs2  44410