Theorem opth2 1909

Description: Equality of the second members of equal ordered pairs. Because of our particular ordered pair definition, equality holds whether or not the first members are sets.

Hypotheses

Ref	Expression
opth2.1	⊢ B ∈ V
opth2.2	⊢ D ∈ V

Assertion

Ref	Expression
opth2	⊢ (⟨A, B⟩ = ⟨C, D⟩ → B = D)

Proof of Theorem opth2

Step	Hyp	Ref	Expression
1		opeq1 1876	. . . . 5 ⊢ (x = A → ⟨x, B⟩ = ⟨A, B⟩)
2	1	cleq1d 1109	. . . 4 ⊢ (x = A → (⟨x, B⟩ = ⟨C, D⟩ ↔ ⟨A, B⟩ = ⟨C, D⟩))
3	2	imbi1d 465	. . 3 ⊢ (x = A → ((⟨x, B⟩ = ⟨C, D⟩ → B = D) ↔ (⟨A, B⟩ = ⟨C, D⟩ → B = D)))
4		visset 1350	. . . . 5 ⊢ x ∈ V
5		opth2.1	. . . . 5 ⊢ B ∈ V
6		opth2.2	. . . . 5 ⊢ D ∈ V
7	4, 5, 6	opth 1898	. . . 4 ⊢ (⟨x, B⟩ = ⟨C, D⟩ ↔ (x = C ∧ B = D))
8	7	pm3.27bd 263	. . 3 ⊢ (⟨x, B⟩ = ⟨C, D⟩ → B = D)
9	3, 8	vtoclg 1383	. 2 ⊢ (A ∈ V → (⟨A, B⟩ = ⟨C, D⟩ → B = D))
10		clneq2 1169	. . . . 5 ⊢ ((∅ ∈ ⟨A, B⟩ ∧ ¬ ∅ ∈ ⟨C, D⟩) → ¬ ⟨A, B⟩ = ⟨C, D⟩)
11		opprc1b 1906	. . . . 5 ⊢ (¬ A ∈ V ↔ ∅ ∈ ⟨A, B⟩)
12		opprc1b 1906	. . . . . . 7 ⊢ (¬ C ∈ V ↔ ∅ ∈ ⟨C, D⟩)
13	12	bicon1i 193	. . . . . 6 ⊢ (¬ ∅ ∈ ⟨C, D⟩ ↔ C ∈ V)
14	13	bicomi 150	. . . . 5 ⊢ (C ∈ V ↔ ¬ ∅ ∈ ⟨C, D⟩)
15	10, 11, 14	syl2anb 350	. . . 4 ⊢ ((¬ A ∈ V ∧ C ∈ V) → ¬ ⟨A, B⟩ = ⟨C, D⟩)
16	15	pm2.21d 74	. . 3 ⊢ ((¬ A ∈ V ∧ C ∈ V) → (⟨A, B⟩ = ⟨C, D⟩ → B = D))
17		opprc1 1905	. . . . 5 ⊢ (¬ A ∈ V → ⟨A, B⟩ = {∅, {B}})
18		opprc1 1905	. . . . 5 ⊢ (¬ C ∈ V → ⟨C, D⟩ = {∅, {D}})
19	17, 18	cleqan12d 1116	. . . 4 ⊢ ((¬ A ∈ V ∧ ¬ C ∈ V) → (⟨A, B⟩ = ⟨C, D⟩ ↔ {∅, {B}} = {∅, {D}}))
20		snex 1859	. . . . . 6 ⊢ {B} ∈ V
21		snex 1859	. . . . . 6 ⊢ {D} ∈ V
22	20, 21	prer2 1873	. . . . 5 ⊢ ({∅, {B}} = {∅, {D}} → {B} = {D})
23	5	sneqr 1856	. . . . 5 ⊢ ({B} = {D} → B = D)
24	22, 23	syl 12	. . . 4 ⊢ ({∅, {B}} = {∅, {D}} → B = D)
25	19, 24	syl6bi 187	. . 3 ⊢ ((¬ A ∈ V ∧ ¬ C ∈ V) → (⟨A, B⟩ = ⟨C, D⟩ → B = D))
26	16, 25	pm2.61an2 365	. 2 ⊢ (¬ A ∈ V → (⟨A, B⟩ = ⟨C, D⟩ → B = D))
27	9, 26	pm2.61i 110	1 ⊢ (⟨A, B⟩ = ⟨C, D⟩ → B = D)

Syntax hints:  ¬ wn 1   → wi 2   ∧ wa 196   = wceq 1091   ∈ wcel 1092  Vcvv 1348  ∅c0 1707  {csn 1808  {cpr 1809  ⟨cop 1810

This theorem is referenced by:  moop2 1910

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

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-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

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