Theorem oneluni 4282

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 4280	. 2 |- ( B e. A -> B C_ A )
3		ssequn2 3188	. 2 |- ( B C_ A <-> ( A u. B ) = A )
4	2, 3	sylib 121	1 |- ( B e. A -> ( A u. B ) = A )

Syntax hints:   -> wi 4   = wceq 1296   e. wcel 1445   u. cun 3011   C_ wss 3013   Oncon0 4214

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077

This theorem depends on definitions:  df-bi 116  df-tru 1299  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ral 2375  df-rex 2376  df-v 2635  df-un 3017  df-in 3019  df-ss 3026  df-uni 3676  df-tr 3959  df-iord 4217  df-on 4219

This theorem is referenced by: (None)