Theorem onelun 2352

Description: An ordinal number equals its union with any element.

Hypothesis

Ref	Expression
on.1	⊢ A ∈ On

Assertion

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

Proof of Theorem onelun

Step	Hyp	Ref	Expression
1		on.1	. . 3 ⊢ A ∈ On
2	1	onelss 2348	. 2 ⊢ (B ∈ A → B ⊆ A)
3		ssequn2 1631	. 2 ⊢ (B ⊆ A ↔ (A ∪ B) = A)
4	2, 3	sylib 173	1 ⊢ (B ∈ A → (A ∪ B) = A)

Syntax hints:   → wi 2   = wceq 1091   ∈ wcel 1092   ∪ cun 1485   ⊆ wss 1487  Oncon0 2199

This theorem is referenced by:  onun 2358

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3an 583  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-tr 2042  df-br 2063  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203

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