Theorem opelopab4 45240

Description: Ordered pair membership in a class abstraction of ordered pairs. Compare to elopab 5513. (Contributed by Alan Sare, 8-Feb-2014.) (Revised by Mario Carneiro, 6-May-2015.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
opelopab4	⊢ (⟨𝑢, 𝑣⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑))

Distinct variable groups:   𝑥,𝑢   𝑦,𝑢   𝑥,𝑣   𝑦,𝑣

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑣,𝑢)

Proof of Theorem opelopab4

Step	Hyp	Ref	Expression
1		elopab 5513	. 2 ⊢ (⟨𝑢, 𝑣⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∃𝑥∃𝑦(⟨𝑢, 𝑣⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
2		vex 3459	. . . . . 6 ⊢ 𝑥 ∈ V
3		vex 3459	. . . . . 6 ⊢ 𝑦 ∈ V
4	2, 3	opth 5460	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ = ⟨𝑢, 𝑣⟩ ↔ (𝑥 = 𝑢 ∧ 𝑦 = 𝑣))
5		eqcom 2770	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ = ⟨𝑢, 𝑣⟩ ↔ ⟨𝑢, 𝑣⟩ = ⟨𝑥, 𝑦⟩)
6	4, 5	bitr3i 280	. . . 4 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ↔ ⟨𝑢, 𝑣⟩ = ⟨𝑥, 𝑦⟩)
7	6	anbi1i 635	. . 3 ⊢ (((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑) ↔ (⟨𝑢, 𝑣⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
8	7	2exbii 1879	. 2 ⊢ (∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑) ↔ ∃𝑥∃𝑦(⟨𝑢, 𝑣⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
9	1, 8	bitr4i 281	1 ⊢ (⟨𝑢, 𝑣⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑))

Syntax hints:   ↔ wb 209   ∧ wa 400   = wceq 1570  ∃wex 1809   ∈ wcel 2143  ⟨cop 4596  {copab 5174

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5258  ax-pr 5406

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-opab 5175

This theorem is referenced by: (None)