Theorem elopab 5500

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 3471	. 2 ⊢ (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} → 𝐴 ∈ V)
2		opex 5431	. . . . 5 ⊢ ⟨𝑥, 𝑦⟩ ∈ V
3		eleq1 2848	. . . . 5 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → (𝐴 ∈ V ↔ ⟨𝑥, 𝑦⟩ ∈ V))
4	2, 3	mpbiri 261	. . . 4 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → 𝐴 ∈ V)
5	4	adantr 486	. . 3 ⊢ ((𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → 𝐴 ∈ V)
6	5	exlimivv 1964	. 2 ⊢ (∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → 𝐴 ∈ V)
7		elopabw 5499	. 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)))
8	1, 6, 7	pm5.21nii 381	1 ⊢ (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   = wceq 1569  ∃wex 1811   ∈ wcel 2144  Vcvv 3450  ⟨cop 4834  {copab 5166

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-ss 3897  df-in 4090  df-un 4150  df-sn 4585  df-pr 4587  df-op 4835  df-opab 5167

This theorem is used by:  rexopabb  5501  vopelopabsb  5502  opelopabsb  5503  opelopabt  5505  opelopabga  5506  brab2d  5511  opabn0  5527  0nelopab  5539  elxp  5673  elopaba  5786  elcnv  5854  cnvopab  6132  dfmpt3  6676  fmptsng  7175  fmptsnd  7176  opabex3d  7979  opabex3rd  7980  opabex3  7981  fsplit  8132  rtrclreclem3  15220  isfunc  18046  dfric2  20764  griedg0ssusgr  29853  rgrusgrprc  30177  brabgaf  33249  qqhval2  34662  eulerpartlemgvv  35056  satfvsucsuc  36218  satf0op  36230  bj-dfres2  38056  elco  38065  opelopabd  38169  opelopabb  38170  poimirlem26  38671  ecxrn  39445  dicelval3  42344  pellexlem5  43947  pellex  43949  opelopab4  45641  sprsymrelfvlem  48679  uspgrsprf  49351  uspgrsprf1  49352  brab2dd  50046