Theorem onun2 4538

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 3791	. 2 |- ( ( A e. On /\ B e. On ) -> { A , B } C_ On )
2		prexg 4255	. . . 4 |- ( ( A e. On /\ B e. On ) -> { A , B } e. _V )
3		ssonuni 4536	. . . 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 3865	. . . 4 |- ( ( A e. On /\ B e. On ) -> U. { A , B } = ( A u. B ) )
6	5	eleq1d 2274	. . 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 2176   _Vcvv 2772   u. cun 3164   C_ wss 3166   {cpr 3634   U.cuni 3850   Oncon0 4410

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pr 4253  ax-un 4480

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-v 2774  df-un 3170  df-in 3172  df-ss 3179  df-sn 3639  df-pr 3640  df-uni 3851  df-tr 4143  df-iord 4413  df-on 4415

This theorem is referenced by:  onun2i  4539  rdgon  6472