Theorem oneluni 6303

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 6299	. 2 ⊢ (𝐵 ∈ 𝐴 → 𝐵 ⊆ 𝐴)
3		ssequn2 4159	. 2 ⊢ (𝐵 ⊆ 𝐴 ↔ (𝐴 ∪ 𝐵) = 𝐴)
4	2, 3	sylib 220	1 ⊢ (𝐵 ∈ 𝐴 → (𝐴 ∪ 𝐵) = 𝐴)

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2114   ∪ cun 3934   ⊆ wss 3936  Oncon0 6191

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-v 3496  df-un 3941  df-in 3943  df-ss 3952  df-uni 4839  df-tr 5173  df-po 5474  df-so 5475  df-fr 5514  df-we 5516  df-ord 6194  df-on 6195

This theorem is referenced by:  onun2i  6306