Theorem preleq 5708

Description: Equality of two unordered pairs when one member of each pair contains the other member.

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 3154	. . . . . 6 |- ({A, B} = {C, D} <-> ((A = C /\ B = D) \/ (A = D /\ B = C)))
6	5	biimpi 168	. . . . 5 |- ({A, B} = {C, D} -> ((A = C /\ B = D) \/ (A = D /\ B = C)))
7	6	ord 249	. . . 4 |- ({A, B} = {C, D} -> (-. (A = C /\ B = D) -> (A = D /\ B = C)))
8		en2lp 5707	. . . . 5 |- -. (D e. C /\ C e. D)
9		eleq12 1959	. . . . . 6 |- ((A = D /\ B = C) -> (A e. B <-> D e. C))
10	9	anbi1d 679	. . . . 5 |- ((A = D /\ B = C) -> ((A e. B /\ C e. D) <-> (D e. C /\ C e. D)))
11	8, 10	mtbiri 785	. . . 4 |- ((A = D /\ B = C) -> -. (A e. B /\ C e. D))
12	7, 11	syl6 25	. . 3 |- ({A, B} = {C, D} -> (-. (A = C /\ B = D) -> -. (A e. B /\ C e. D)))
13	12	con4d 91	. 2 |- ({A, B} = {C, D} -> ((A e. B /\ C e. D) -> (A = C /\ B = D)))
14	13	impcom 378	1 |- (((A e. B /\ C e. D) /\ {A, B} = {C, D}) -> (A = C /\ B = D))

Syntax hints:  -. wn 2   -> wi 3   \/ wo 239   /\ wa 240   = wceq 1298   e. wcel 1300  _Vcvv 2292  {cpr 3045

This theorem is referenced by:  opthreg 5709  aceq6b 5904

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-reg 5695

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-br 3339  df-opab 3396  df-eprel 3583  df-fr 3625

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