Theorem preleq 3454

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

Hypotheses

Ref	Expression
preleq.1	⊢ A ∈ V
preleq.2	⊢ B ∈ V
preleq.3	⊢ C ∈ V
preleq.4	⊢ D ∈ V

Assertion

Ref	Expression
preleq	⊢ (((A ∈ B ∧ C ∈ D) ∧ {A, B} = {C, D}) → (A = C ∧ B = D))

Proof of Theorem preleq

Step	Hyp	Ref	Expression
1		preleq.1	. . . . . . . 8 ⊢ A ∈ V
2		preleq.2	. . . . . . . 8 ⊢ B ∈ V
3		preleq.3	. . . . . . . 8 ⊢ C ∈ V
4		preleq.4	. . . . . . . 8 ⊢ D ∈ V
5	1, 2, 3, 4	preq12b 1874	. . . . . . 7 ⊢ ({A, B} = {C, D} ↔ ((A = C ∧ B = D) ∨ (A = D ∧ B = C)))
6	5	biimp 133	. . . . . 6 ⊢ ({A, B} = {C, D} → ((A = C ∧ B = D) ∨ (A = D ∧ B = C)))
7	6	ord 202	. . . . 5 ⊢ ({A, B} = {C, D} → (¬ (A = C ∧ B = D) → (A = D ∧ B = C)))
8		en2lp 3453	. . . . . 6 ⊢ ¬ (D ∈ C ∧ C ∈ D)
9		eleq12 1151	. . . . . . 7 ⊢ ((A = D ∧ B = C) → (A ∈ B ↔ D ∈ C))
10	9	anbi1d 469	. . . . . 6 ⊢ ((A = D ∧ B = C) → ((A ∈ B ∧ C ∈ D) ↔ (D ∈ C ∧ C ∈ D)))
11	8, 10	mtbiri 539	. . . . 5 ⊢ ((A = D ∧ B = C) → ¬ (A ∈ B ∧ C ∈ D))
12	7, 11	syl6 23	. . . 4 ⊢ ({A, B} = {C, D} → (¬ (A = C ∧ B = D) → ¬ (A ∈ B ∧ C ∈ D)))
13	12	a3d 70	. . 3 ⊢ ({A, B} = {C, D} → ((A ∈ B ∧ C ∈ D) → (A = C ∧ B = D)))
14	13	imp 277	. 2 ⊢ (({A, B} = {C, D} ∧ (A ∈ B ∧ C ∈ D)) → (A = C ∧ B = D))
15	14	ancoms 334	1 ⊢ (((A ∈ B ∧ C ∈ D) ∧ {A, B} = {C, D}) → (A = C ∧ B = D))

Syntax hints:  ¬ wn 1   → wi 2   ∨ wo 195   ∧ wa 196   = wceq 1091   ∈ wcel 1092  Vcvv 1348  {cpr 1809

This theorem is referenced by:  opthreg 3455  aceq6b 3565

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077  ax-reg 1078

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-br 2063  df-opab 2098  df-eprel 2122  df-fr 2169

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