Theorem ordequn 5828

Description: The maximum (i.e. union) of two ordinals is either one or the other. Similar to Exercise 14 of [TakeutiZaring] p. 40. (Contributed by NM, 28-Nov-2003.)

Assertion

Ref	Expression
ordequn	|- ( ( Ord B /\ Ord C ) -> ( A = ( B u. C ) -> ( A = B \/ A = C ) ) )

Proof of Theorem ordequn

Step	Hyp	Ref	Expression
1		ordtri2or2 5823	. . 3 |- ( ( Ord B /\ Ord C ) -> ( B C_ C \/ C C_ B ) )
2	1	orcomd 403	. 2 |- ( ( Ord B /\ Ord C ) -> ( C C_ B \/ B C_ C ) )
3		eqeq1 2626	. . . 4 |- ( A = ( B u. C ) -> ( A = B <-> ( B u. C ) = B ) )
4		ssequn2 3786	. . . 4 |- ( C C_ B <-> ( B u. C ) = B )
5	3, 4	syl6rbbr 279	. . 3 |- ( A = ( B u. C ) -> ( C C_ B <-> A = B ) )
6		eqeq1 2626	. . . 4 |- ( A = ( B u. C ) -> ( A = C <-> ( B u. C ) = C ) )
7		ssequn1 3783	. . . 4 |- ( B C_ C <-> ( B u. C ) = C )
8	6, 7	syl6rbbr 279	. . 3 |- ( A = ( B u. C ) -> ( B C_ C <-> A = C ) )
9	5, 8	orbi12d 746	. 2 |- ( A = ( B u. C ) -> ( ( C C_ B \/ B C_ C ) <-> ( A = B \/ A = C ) ) )
10	2, 9	syl5ibcom 235	1 |- ( ( Ord B /\ Ord C ) -> ( A = ( B u. C ) -> ( A = B \/ A = C ) ) )

Syntax hints:   -> wi 4   \/ wo 383   /\ wa 384   = wceq 1483   u. cun 3572   C_ wss 3574   Ord word 5722

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-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

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

This theorem is referenced by:  ordun  5829  inar1  9597