Theorem ordunpr 7026

Description: The maximum of two ordinals is equal to one of them. (Contributed by Mario Carneiro, 25-Jun-2015.)

Assertion

Ref	Expression
ordunpr	|- ( ( B e. On /\ C e. On ) -> ( B u. C ) e. { B , C } )

Proof of Theorem ordunpr

Step	Hyp	Ref	Expression
1		eloni 5733	. . . . 5 |- ( B e. On -> Ord B )
2		eloni 5733	. . . . 5 |- ( C e. On -> Ord C )
3		ordtri2or2 5823	. . . . 5 |- ( ( Ord B /\ Ord C ) -> ( B C_ C \/ C C_ B ) )
4	1, 2, 3	syl2an 494	. . . 4 |- ( ( B e. On /\ C e. On ) -> ( B C_ C \/ C C_ B ) )
5	4	orcomd 403	. . 3 |- ( ( B e. On /\ C e. On ) -> ( C C_ B \/ B C_ C ) )
6		ssequn2 3786	. . . 4 |- ( C C_ B <-> ( B u. C ) = B )
7		ssequn1 3783	. . . 4 |- ( B C_ C <-> ( B u. C ) = C )
8	6, 7	orbi12i 543	. . 3 |- ( ( C C_ B \/ B C_ C ) <-> ( ( B u. C ) = B \/ ( B u. C ) = C ) )
9	5, 8	sylib 208	. 2 |- ( ( B e. On /\ C e. On ) -> ( ( B u. C ) = B \/ ( B u. C ) = C ) )
10		unexg 6959	. . 3 |- ( ( B e. On /\ C e. On ) -> ( B u. C ) e. _V )
11		elprg 4196	. . 3 |- ( ( B u. C ) e. _V -> ( ( B u. C ) e. { B , C } <-> ( ( B u. C ) = B \/ ( B u. C ) = C ) ) )
12	10, 11	syl 17	. 2 |- ( ( B e. On /\ C e. On ) -> ( ( B u. C ) e. { B , C } <-> ( ( B u. C ) = B \/ ( B u. C ) = C ) ) )
13	9, 12	mpbird 247	1 |- ( ( B e. On /\ C e. On ) -> ( B u. C ) e. { B , C } )

Syntax hints:   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   u. cun 3572   C_ wss 3574   {cpr 4179   Ord word 5722   Oncon0 5723

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-tr 4753  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-ord 5726  df-on 5727

This theorem is referenced by:  ordunel  7027  r0weon  8835