Theorem preleq 8514

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		preleq.1	. . . . . . 7 |- A e. _V
2		preleq.2	. . . . . . 7 |- B e. _V
3		preleq.3	. . . . . . 7 |- C e. _V
4		preleq.4	. . . . . . 7 |- D e. _V
5	1, 2, 3, 4	preq12b 4382	. . . . . 6 |- ( { A , B } = { C , D } <-> ( ( A = C /\ B = D ) \/ ( A = D /\ B = C ) ) )
6	5	biimpi 206	. . . . 5 |- ( { A , B } = { C , D } -> ( ( A = C /\ B = D ) \/ ( A = D /\ B = C ) ) )
7	6	ord 392	. . . 4 |- ( { A , B } = { C , D } -> ( -. ( A = C /\ B = D ) -> ( A = D /\ B = C ) ) )
8		en2lp 8510	. . . . 5 |- -. ( D e. C /\ C e. D )
9		eleq12 2691	. . . . . 6 |- ( ( A = D /\ B = C ) -> ( A e. B <-> D e. C ) )
10	9	anbi1d 741	. . . . 5 |- ( ( A = D /\ B = C ) -> ( ( A e. B /\ C e. D ) <-> ( D e. C /\ C e. D ) ) )
11	8, 10	mtbiri 317	. . . 4 |- ( ( A = D /\ B = C ) -> -. ( A e. B /\ C e. D ) )
12	7, 11	syl6 35	. . 3 |- ( { A , B } = { C , D } -> ( -. ( A = C /\ B = D ) -> -. ( A e. B /\ C e. D ) ) )
13	12	con4d 114	. 2 |- ( { A , B } = { C , D } -> ( ( A e. B /\ C e. D ) -> ( A = C /\ B = D ) ) )
14	13	impcom 446	1 |- ( ( ( A e. B /\ C e. D ) /\ { A , B } = { C , D } ) -> ( A = C /\ B = D ) )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   {cpr 4179

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-reg 8497

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-eprel 5029  df-fr 5073

This theorem is referenced by:  opthreg  8515  dfac2  8953