Theorem brdifun 6156

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 4394	. . . 4 |- ( ( A e. X /\ B e. X ) -> <. A , B >. e. ( X X. X ) )
2		df-br 3786	. . . 4 |- ( A ( X X. X ) B <-> <. A , B >. e. ( X X. X ) )
3	1, 2	sylibr 132	. . 3 |- ( ( A e. X /\ B e. X ) -> A ( X X. X ) B )
4		swoer.1	. . . . . 6 |- R = ( ( X X. X ) \ ( .< u. `' .< ) )
5	4	breqi 3791	. . . . 5 |- ( A R B <-> A ( ( X X. X ) \ ( .< u. `' .< ) ) B )
6		brdif 3833	. . . . 5 |- ( A ( ( X X. X ) \ ( .< u. `' .< ) ) B <-> ( A ( X X. X ) B /\ -. A ( .< u. `' .< ) B ) )
7	5, 6	bitri 182	. . . 4 |- ( A R B <-> ( A ( X X. X ) B /\ -. A ( .< u. `' .< ) B ) )
8	7	baib 861	. . 3 |- ( A ( X X. X ) B -> ( A R B <-> -. A ( .< u. `' .< ) B ) )
9	3, 8	syl 14	. 2 |- ( ( A e. X /\ B e. X ) -> ( A R B <-> -. A ( .< u. `' .< ) B ) )
10		brun 3831	. . . 4 |- ( A ( .< u. `' .< ) B <-> ( A .< B \/ A `' .< B ) )
11		brcnvg 4534	. . . . 5 |- ( ( A e. X /\ B e. X ) -> ( A `' .< B <-> B .< A ) )
12	11	orbi2d 736	. . . 4 |- ( ( A e. X /\ B e. X ) -> ( ( A .< B \/ A `' .< B ) <-> ( A .< B \/ B .< A ) ) )
13	10, 12	syl5bb 190	. . 3 |- ( ( A e. X /\ B e. X ) -> ( A ( .< u. `' .< ) B <-> ( A .< B \/ B .< A ) ) )
14	13	notbid 624	. 2 |- ( ( A e. X /\ B e. X ) -> ( -. A ( .< u. `' .< ) B <-> -. ( A .< B \/ B .< A ) ) )
15	9, 14	bitrd 186	1 |- ( ( A e. X /\ B e. X ) -> ( A R B <-> -. ( A .< B \/ B .< A ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 102   <-> wb 103   \/ wo 661   = wceq 1284   e. wcel 1433   \ cdif 2970   u. cun 2971   <.cop 3401   class class class wbr 3785   X. cxp 4361   `'ccnv 4362

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-br 3786  df-opab 3840  df-xp 4369  df-cnv 4371

This theorem is referenced by:  swoer  6157  swoord1  6158  swoord2  6159