Theorem opelopab4 42060

Description: Ordered pair membership in a class abstraction of ordered pairs. Compare to elopab 5433. (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 5433	. 2 ⊢ (⟨𝑢, 𝑣⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∃𝑥∃𝑦(⟨𝑢, 𝑣⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
2		vex 3426	. . . . . 6 ⊢ 𝑥 ∈ V
3		vex 3426	. . . . . 6 ⊢ 𝑦 ∈ V
4	2, 3	opth 5385	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ = ⟨𝑢, 𝑣⟩ ↔ (𝑥 = 𝑢 ∧ 𝑦 = 𝑣))
5		eqcom 2745	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ = ⟨𝑢, 𝑣⟩ ↔ ⟨𝑢, 𝑣⟩ = ⟨𝑥, 𝑦⟩)
6	4, 5	bitr3i 276	. . . 4 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ↔ ⟨𝑢, 𝑣⟩ = ⟨𝑥, 𝑦⟩)
7	6	anbi1i 623	. . 3 ⊢ (((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑) ↔ (⟨𝑢, 𝑣⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
8	7	2exbii 1852	. 2 ⊢ (∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑) ↔ ∃𝑥∃𝑦(⟨𝑢, 𝑣⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
9	1, 8	bitr4i 277	1 ⊢ (⟨𝑢, 𝑣⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑))

Syntax hints:   ↔ wb 205   ∧ wa 395   = wceq 1539  ∃wex 1783   ∈ wcel 2108  ⟨cop 4564  {copab 5132

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-opab 5133

This theorem is referenced by: (None)