Theorem elopab 5482

Description: Membership in a class abstraction of ordered pairs. (Contributed by NM, 24-Mar-1998.)

Assertion

Ref	Expression
elopab	⊢ (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))

Distinct variable groups:   𝑥,𝐴   𝑦,𝐴

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem elopab

Step	Hyp	Ref	Expression
1		elex 3465	. 2 ⊢ (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} → 𝐴 ∈ V)
2		opex 5419	. . . . 5 ⊢ ⟨𝑥, 𝑦⟩ ∈ V
3		eleq1 2816	. . . . 5 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → (𝐴 ∈ V ↔ ⟨𝑥, 𝑦⟩ ∈ V))
4	2, 3	mpbiri 258	. . . 4 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → 𝐴 ∈ V)
5	4	adantr 480	. . 3 ⊢ ((𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → 𝐴 ∈ V)
6	5	exlimivv 1932	. 2 ⊢ (∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → 𝐴 ∈ V)
7		elopabw 5481	. 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)))
8	1, 6, 7	pm5.21nii 378	1 ⊢ (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2109  Vcvv 3444  ⟨cop 4591  {copab 5164

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pr 5382

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-v 3446  df-dif 3914  df-un 3916  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-opab 5165

This theorem is referenced by:  rexopabb  5483  vopelopabsb  5484  opelopabsb  5485  opelopabt  5487  opelopabga  5488  opabn0  5508  0nelopab  5520  elxp  5654  elopaba  5762  elcnv  5830  cnvopab  6098  dfmpt3  6634  fmptsng  7124  fmptsnd  7125  opabex3d  7923  opabex3rd  7924  opabex3  7925  fsplit  8073  rtrclreclem3  15002  isfunc  17802  griedg0ssusgr  29168  rgrusgrprc  29493  brab2d  32508  brabgaf  32509  qqhval2  33945  eulerpartlemgvv  34340  satfvsucsuc  35325  satf0op  35337  opelopabd  37102  opelopabb  37103  poimirlem26  37613  ecxrn  38346  dicelval3  41147  pellexlem5  42794  pellex  42796  opelopab4  44514  sprsymrelfvlem  47464  uspgrsprf  48107  uspgrsprf1  48108  brab2dd  48789