Theorem brdifun 7771

Description: Evaluate the incomparability relation. (Contributed by Mario Carneiro, 9-Jul-2014.)

Hypothesis

Ref	Expression
swoer.1	|- R = ( ( X X. X ) \ ( .< u. `' .< ) )

Assertion

Ref	Expression
brdifun	|- ( ( A e. X /\ B e. X ) -> ( A R B <-> -. ( A .< B \/ B .< A ) ) )

Proof of Theorem brdifun

Step	Hyp	Ref	Expression
1		opelxpi 5148	. . . 4 |- ( ( A e. X /\ B e. X ) -> <. A , B >. e. ( X X. X ) )
2		df-br 4654	. . . 4 |- ( A ( X X. X ) B <-> <. A , B >. e. ( X X. X ) )
3	1, 2	sylibr 224	. . 3 |- ( ( A e. X /\ B e. X ) -> A ( X X. X ) B )
4		swoer.1	. . . . . 6 |- R = ( ( X X. X ) \ ( .< u. `' .< ) )
5	4	breqi 4659	. . . . 5 |- ( A R B <-> A ( ( X X. X ) \ ( .< u. `' .< ) ) B )
6		brdif 4705	. . . . 5 |- ( A ( ( X X. X ) \ ( .< u. `' .< ) ) B <-> ( A ( X X. X ) B /\ -. A ( .< u. `' .< ) B ) )
7	5, 6	bitri 264	. . . 4 |- ( A R B <-> ( A ( X X. X ) B /\ -. A ( .< u. `' .< ) B ) )
8	7	baib 944	. . 3 |- ( A ( X X. X ) B -> ( A R B <-> -. A ( .< u. `' .< ) B ) )
9	3, 8	syl 17	. 2 |- ( ( A e. X /\ B e. X ) -> ( A R B <-> -. A ( .< u. `' .< ) B ) )
10		brun 4703	. . . 4 |- ( A ( .< u. `' .< ) B <-> ( A .< B \/ A `' .< B ) )
11		brcnvg 5303	. . . . 5 |- ( ( A e. X /\ B e. X ) -> ( A `' .< B <-> B .< A ) )
12	11	orbi2d 738	. . . 4 |- ( ( A e. X /\ B e. X ) -> ( ( A .< B \/ A `' .< B ) <-> ( A .< B \/ B .< A ) ) )
13	10, 12	syl5bb 272	. . 3 |- ( ( A e. X /\ B e. X ) -> ( A ( .< u. `' .< ) B <-> ( A .< B \/ B .< A ) ) )
14	13	notbid 308	. 2 |- ( ( A e. X /\ B e. X ) -> ( -. A ( .< u. `' .< ) B <-> -. ( A .< B \/ B .< A ) ) )
15	9, 14	bitrd 268	1 |- ( ( A e. X /\ B e. X ) -> ( A R B <-> -. ( A .< B \/ B .< A ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   \ cdif 3571   u. cun 3572   <.cop 4183   class class class wbr 4653   X. cxp 5112   `'ccnv 5113

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-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-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-xp 5120  df-cnv 5122

This theorem is referenced by:  swoer  7772  swoord1  7773  swoord2  7774  swoso  7775