Theorem preleq 4532

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 4531	. . . . 5 |- -. ( D e. C /\ C e. D )
2		eleq12 2231	. . . . . 6 |- ( ( A = D /\ B = C ) -> ( A e. B <-> D e. C ) )
3	2	anbi1d 461	. . . . 5 |- ( ( A = D /\ B = C ) -> ( ( A e. B /\ C e. D ) <-> ( D e. C /\ C e. D ) ) )
4	1, 3	mtbiri 665	. . . 4 |- ( ( A = D /\ B = C ) -> -. ( A e. B /\ C e. D ) )
5	4	con2i 617	. . 3 |- ( ( A e. B /\ C e. D ) -> -. ( A = D /\ B = C ) )
6	5	adantr 274	. 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 3750	. . . 4 |- ( { A , B } = { C , D } <-> ( ( A = C /\ B = D ) \/ ( A = D /\ B = C ) ) )
12	11	biimpi 119	. . 3 |- ( { A , B } = { C , D } -> ( ( A = C /\ B = D ) \/ ( A = D /\ B = C ) ) )
13	12	adantl 275	. 2 |- ( ( ( A e. B /\ C e. D ) /\ { A , B } = { C , D } ) -> ( ( A = C /\ B = D ) \/ ( A = D /\ B = C ) ) )
14	6, 13	ecased 1339	1 |- ( ( ( A e. B /\ C e. D ) /\ { A , B } = { C , D } ) -> ( A = C /\ B = D ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   \/ wo 698   = wceq 1343   e. wcel 2136   _Vcvv 2726   {cpr 3577

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147  ax-setind 4514

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-v 2728  df-dif 3118  df-un 3120  df-sn 3582  df-pr 3583

This theorem is referenced by:  opthreg  4533