Theorem brdifun 6772

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 4763	. . . 4 |- ( ( A e. X /\ B e. X ) -> <. A , B >. e. ( X X. X ) )
2		df-br 4094	. . . 4 |- ( A ( X X. X ) B <-> <. A , B >. e. ( X X. X ) )
3	1, 2	sylibr 134	. . 3 |- ( ( A e. X /\ B e. X ) -> A ( X X. X ) B )
4		swoer.1	. . . . . 6 |- R = ( ( X X. X ) \ ( .< u. `' .< ) )
5	4	breqi 4099	. . . . 5 |- ( A R B <-> A ( ( X X. X ) \ ( .< u. `' .< ) ) B )
6		brdif 4147	. . . . 5 |- ( A ( ( X X. X ) \ ( .< u. `' .< ) ) B <-> ( A ( X X. X ) B /\ -. A ( .< u. `' .< ) B ) )
7	5, 6	bitri 184	. . . 4 |- ( A R B <-> ( A ( X X. X ) B /\ -. A ( .< u. `' .< ) B ) )
8	7	baib 927	. . 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 4145	. . . 4 |- ( A ( .< u. `' .< ) B <-> ( A .< B \/ A `' .< B ) )
11		brcnvg 4917	. . . . 5 |- ( ( A e. X /\ B e. X ) -> ( A `' .< B <-> B .< A ) )
12	11	orbi2d 798	. . . 4 |- ( ( A e. X /\ B e. X ) -> ( ( A .< B \/ A `' .< B ) <-> ( A .< B \/ B .< A ) ) )
13	10, 12	bitrid 192	. . 3 |- ( ( A e. X /\ B e. X ) -> ( A ( .< u. `' .< ) B <-> ( A .< B \/ B .< A ) ) )
14	13	notbid 673	. 2 |- ( ( A e. X /\ B e. X ) -> ( -. A ( .< u. `' .< ) B <-> -. ( A .< B \/ B .< A ) ) )
15	9, 14	bitrd 188	1 |- ( ( A e. X /\ B e. X ) -> ( A R B <-> -. ( A .< B \/ B .< A ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 716   = wceq 1398   e. wcel 2202   \ cdif 3198   u. cun 3199   <.cop 3676   class class class wbr 4093   X. cxp 4729   `'ccnv 4730

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-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-br 4094  df-opab 4156  df-xp 4737  df-cnv 4739

This theorem is referenced by:  swoer  6773  swoord1  6774  swoord2  6775