Theorem preleq 4647

Description: Equality of two unordered pairs when one member of each pair contains the other member. (Contributed by NM, 16-Oct-1996.)

Hypotheses

Ref	Expression
preleq.1	|- A e. _V
preleq.2	|- B e. _V
preleq.3	|- C e. _V
preleq.4	|- D e. _V

Assertion

Ref	Expression
preleq	|- ( ( ( A e. B /\ C e. D ) /\ { A , B } = { C , D } ) -> ( A = C /\ B = D ) )

Proof of Theorem preleq

Step	Hyp	Ref	Expression
1		en2lp 4646	. . . . 5 |- -. ( D e. C /\ C e. D )
2		eleq12 2294	. . . . . 6 |- ( ( A = D /\ B = C ) -> ( A e. B <-> D e. C ) )
3	2	anbi1d 465	. . . . 5 |- ( ( A = D /\ B = C ) -> ( ( A e. B /\ C e. D ) <-> ( D e. C /\ C e. D ) ) )
4	1, 3	mtbiri 679	. . . 4 |- ( ( A = D /\ B = C ) -> -. ( A e. B /\ C e. D ) )
5	4	con2i 630	. . 3 |- ( ( A e. B /\ C e. D ) -> -. ( A = D /\ B = C ) )
6	5	adantr 276	. 2 |- ( ( ( A e. B /\ C e. D ) /\ { A , B } = { C , D } ) -> -. ( A = D /\ B = C ) )
7		preleq.1	. . . . 5 |- A e. _V
8		preleq.2	. . . . 5 |- B e. _V
9		preleq.3	. . . . 5 |- C e. _V
10		preleq.4	. . . . 5 |- D e. _V
11	7, 8, 9, 10	preq12b 3848	. . . 4 |- ( { A , B } = { C , D } <-> ( ( A = C /\ B = D ) \/ ( A = D /\ B = C ) ) )
12	11	biimpi 120	. . 3 |- ( { A , B } = { C , D } -> ( ( A = C /\ B = D ) \/ ( A = D /\ B = C ) ) )
13	12	adantl 277	. 2 |- ( ( ( A e. B /\ C e. D ) /\ { A , B } = { C , D } ) -> ( ( A = C /\ B = D ) \/ ( A = D /\ B = C ) ) )
14	6, 13	ecased 1383	1 |- ( ( ( A e. B /\ C e. D ) /\ { A , B } = { C , D } ) -> ( A = C /\ B = D ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   \/ wo 713   = wceq 1395   e. wcel 2200   _Vcvv 2799   {cpr 3667

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211  ax-setind 4629

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-v 2801  df-dif 3199  df-un 3201  df-sn 3672  df-pr 3673

This theorem is referenced by:  opthreg  4648