Theorem preleq 4603

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 4602	. . . . 5 |- -. ( D e. C /\ C e. D )
2		eleq12 2270	. . . . . 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 677	. . . 4 |- ( ( A = D /\ B = C ) -> -. ( A e. B /\ C e. D ) )
5	4	con2i 628	. . 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 3811	. . . 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 1362	1 |- ( ( ( A e. B /\ C e. D ) /\ { A , B } = { C , D } ) -> ( A = C /\ B = D ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   \/ wo 710   = wceq 1373   e. wcel 2176   _Vcvv 2772   {cpr 3634

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 615  ax-in2 616  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187  ax-setind 4585

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-v 2774  df-dif 3168  df-un 3170  df-sn 3639  df-pr 3640

This theorem is referenced by:  opthreg  4604