Theorem onun2 4307

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 3595	. 2 |- ( ( A e. On /\ B e. On ) -> { A , B } C_ On )
2		prexg 4038	. . . 4 |- ( ( A e. On /\ B e. On ) -> { A , B } e. _V )
3		ssonuni 4305	. . . 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 3668	. . . 4 |- ( ( A e. On /\ B e. On ) -> U. { A , B } = ( A u. B ) )
6	5	eleq1d 2156	. . 3 |- ( ( A e. On /\ B e. On ) -> ( U. { A , B } e. On <-> ( A u. B ) e. On ) )
7	4, 6	sylibd 147	. 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 102   e. wcel 1438   _Vcvv 2619   u. cun 2997   C_ wss 2999   {cpr 3447   U.cuni 3653   Oncon0 4190

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pr 4036  ax-un 4260

This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-sn 3452  df-pr 3453  df-uni 3654  df-tr 3937  df-iord 4193  df-on 4195

This theorem is referenced by:  onun2i  4308  rdgon  6151