Theorem preleq 4279

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 4278	. . . . 5 |- -. ( D e. C /\ C e. D )
2		eleq12 2102	. . . . . 6 |- ( ( A = D /\ B = C ) -> ( A e. B <-> D e. C ) )
3	2	anbi1d 438	. . . . 5 |- ( ( A = D /\ B = C ) -> ( ( A e. B /\ C e. D ) <-> ( D e. C /\ C e. D ) ) )
4	1, 3	mtbiri 600	. . . 4 |- ( ( A = D /\ B = C ) -> -. ( A e. B /\ C e. D ) )
5	4	con2i 557	. . 3 |- ( ( A e. B /\ C e. D ) -> -. ( A = D /\ B = C ) )
6	5	adantr 261	. 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 3541	. . . 4 |- ( { A , B } = { C , D } <-> ( ( A = C /\ B = D ) \/ ( A = D /\ B = C ) ) )
12	11	biimpi 113	. . 3 |- ( { A , B } = { C , D } -> ( ( A = C /\ B = D ) \/ ( A = D /\ B = C ) ) )
13	12	adantl 262	. 2 |- ( ( ( A e. B /\ C e. D ) /\ { A , B } = { C , D } ) -> ( ( A = C /\ B = D ) \/ ( A = D /\ B = C ) ) )
14	6, 13	ecased 1239	1 |- ( ( ( A e. B /\ C e. D ) /\ { A , B } = { C , D } ) -> ( A = C /\ B = D ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 97   \/ wo 629   = wceq 1243   e. wcel 1393   _Vcvv 2557   {cpr 3376

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-setind 4262

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-v 2559  df-dif 2920  df-un 2922  df-sn 3381  df-pr 3382

This theorem is referenced by:  opthreg  4280