Theorem oneluni 6482

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 6478	. 2 ⊢ (𝐵 ∈ 𝐴 → 𝐵 ⊆ 𝐴)
3		ssequn2 4138	. 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 1570   ∈ wcel 2145   ∪ cun 3900   ⊆ wss 3902  Oncon0 6361

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3079  df-v 3455  df-un 3907  df-ss 3919  df-uni 4871  df-tr 5217  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-ord 6364  df-on 6365

This theorem is used by:  omabs2  44175