Theorem onun2 4488

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 3750	. 2 |- ( ( A e. On /\ B e. On ) -> { A , B } C_ On )
2		prexg 4210	. . . 4 |- ( ( A e. On /\ B e. On ) -> { A , B } e. _V )
3		ssonuni 4486	. . . 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 3824	. . . 4 |- ( ( A e. On /\ B e. On ) -> U. { A , B } = ( A u. B ) )
6	5	eleq1d 2246	. . 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 2148   _Vcvv 2737   u. cun 3127   C_ wss 3129   {cpr 3593   U.cuni 3809   Oncon0 4362

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-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4120  ax-pr 4208  ax-un 4432

This theorem depends on definitions:  df-bi 117  df-3an 980  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 2739  df-un 3133  df-in 3135  df-ss 3142  df-sn 3598  df-pr 3599  df-uni 3810  df-tr 4101  df-iord 4365  df-on 4367

This theorem is referenced by:  onun2i  4489  rdgon  6383