Theorem oneluni 4433

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 4431	. 2 |- ( B e. A -> B C_ A )
3		ssequn2 3310	. 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 1353   e. wcel 2148   u. cun 3129   C_ wss 3131   Oncon0 4365

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-un 3135  df-in 3137  df-ss 3144  df-uni 3812  df-tr 4104  df-iord 4368  df-on 4370

This theorem is referenced by: (None)