Theorem onun2 4467

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 3731	. 2 |- ( ( A e. On /\ B e. On ) -> { A , B } C_ On )
2		prexg 4189	. . . 4 |- ( ( A e. On /\ B e. On ) -> { A , B } e. _V )
3		ssonuni 4465	. . . 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 3804	. . . 4 |- ( ( A e. On /\ B e. On ) -> U. { A , B } = ( A u. B ) )
6	5	eleq1d 2235	. . 3 |- ( ( A e. On /\ B e. On ) -> ( U. { A , B } e. On <-> ( A u. B ) e. On ) )
7	4, 6	sylibd 148	. 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 103   e. wcel 2136   _Vcvv 2726   u. cun 3114   C_ wss 3116   {cpr 3577   U.cuni 3789   Oncon0 4341

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pr 4187  ax-un 4411

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-sn 3582  df-pr 3583  df-uni 3790  df-tr 4081  df-iord 4344  df-on 4346

This theorem is referenced by:  onun2i  4468  rdgon  6354