Theorem preleq 4583

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 2479	. . . . . 6 |- ({A, B} = {C, D} <-> ((A = C /\ B = D) \/ (A = D /\ B = C)))
6	5	biimp 151	. . . . 5 |- ({A, B} = {C, D} -> ((A = C /\ B = D) \/ (A = D /\ B = C)))
7	6	ord 232	. . . 4 |- ({A, B} = {C, D} -> (-. (A = C /\ B = D) -> (A = D /\ B = C)))
8		en2lp 4582	. . . . 5 |- -. (D e. C /\ C e. D)
9		eleq12 1533	. . . . . 6 |- ((A = D /\ B = C) -> (A e. B <-> D e. C))
10	9	anbi1d 616	. . . . 5 |- ((A = D /\ B = C) -> ((A e. B /\ C e. D) <-> (D e. C /\ C e. D)))
11	8, 10	mtbiri 716	. . . 4 |- ((A = D /\ B = C) -> -. (A e. B /\ C e. D))
12	7, 11	syl6 22	. . 3 |- ({A, B} = {C, D} -> (-. (A = C /\ B = D) -> -. (A e. B /\ C e. D)))
13	12	a3d 75	. 2 |- ({A, B} = {C, D} -> ((A e. B /\ C e. D) -> (A = C /\ B = D)))
14	13	impcom 351	1 |- (((A e. B /\ C e. D) /\ {A, B} = {C, D}) -> (A = C /\ B = D))

Syntax hints:  -. wn 2   -> wi 3   \/ wo 222   /\ wa 223   = wceq 954   e. wcel 956  Vcvv 1807  {cpr 2406

This theorem is referenced by:  opthreg 4584  aceq6b 4722

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-sep 2698  ax-pow 2737  ax-pr 2774  ax-reg 4573

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 776  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-ral 1646  df-rex 1647  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-op 2412  df-br 2615  df-opab 2662  df-eprel 2827  df-fr 2912

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