Theorem oneluni 4446

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

Hypothesis

Ref	Expression
on.1	|- A e. On

Assertion

Ref	Expression
oneluni	|- ( B e. A -> ( A u. B ) = A )

Proof of Theorem oneluni

Step	Hyp	Ref	Expression
1		on.1	. . 3 |- A e. On
2	1	onelssi 4444	. 2 |- ( B e. A -> B C_ A )
3		ssequn2 3323	. 2 |- ( B C_ A <-> ( A u. B ) = A )
4	2, 3	sylib 122	1 |- ( B e. A -> ( A u. B ) = A )

Syntax hints:   -> wi 4   = wceq 1364   e. wcel 2160   u. cun 3142   C_ wss 3144   Oncon0 4378

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-uni 3825  df-tr 4117  df-iord 4381  df-on 4383

This theorem is referenced by: (None)