Theorem onun2 4617

Description: The union of two ordinal numbers is an ordinal number. (Contributed by Jim Kingdon, 25-Jul-2019.)

Assertion

Ref	Expression
onun2	|- ( ( A e. On /\ B e. On ) -> ( A u. B ) e. On )

Proof of Theorem onun2

Step	Hyp	Ref	Expression
1		prssi 3857	. 2 |- ( ( A e. On /\ B e. On ) -> { A , B } C_ On )
2		prexg 4330	. . . 4 |- ( ( A e. On /\ B e. On ) -> { A , B } e. _V )
3		ssonuni 4615	. . . 4 |- ( { A , B } e. _V -> ( { A , B } C_ On -> U. { A , B } e. On ) )
4	2, 3	syl 14	. . 3 |- ( ( A e. On /\ B e. On ) -> ( { A , B } C_ On -> U. { A , B } e. On ) )
5		uniprg 3934	. . . 4 |- ( ( A e. On /\ B e. On ) -> U. { A , B } = ( A u. B ) )
6	5	eleq1d 2303	. . 3 |- ( ( A e. On /\ B e. On ) -> ( U. { A , B } e. On <-> ( A u. B ) e. On ) )
7	4, 6	sylibd 149	. 2 |- ( ( A e. On /\ B e. On ) -> ( { A , B } C_ On -> ( A u. B ) e. On ) )
8	1, 7	mpd 13	1 |- ( ( A e. On /\ B e. On ) -> ( A u. B ) e. On )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2205   _Vcvv 2815   u. cun 3212   C_ wss 3214   {cpr 3695   U.cuni 3919   Oncon0 4489

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pr 4327  ax-un 4559

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-un 3218  df-in 3220  df-ss 3227  df-sn 3700  df-pr 3701  df-uni 3920  df-tr 4214  df-iord 4492  df-on 4494

This theorem is referenced by:  onun2i  4618  rdgon  6630