Theorem elopab 4243

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

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 2741	. 2 ⊢ (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} → 𝐴 ∈ V)
2		vex 2733	. . . . . 6 ⊢ 𝑥 ∈ V
3		vex 2733	. . . . . 6 ⊢ 𝑦 ∈ V
4	2, 3	opex 4214	. . . . 5 ⊢ ⟨𝑥, 𝑦⟩ ∈ V
5		eleq1 2233	. . . . 5 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → (𝐴 ∈ V ↔ ⟨𝑥, 𝑦⟩ ∈ V))
6	4, 5	mpbiri 167	. . . 4 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → 𝐴 ∈ V)
7	6	adantr 274	. . 3 ⊢ ((𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → 𝐴 ∈ V)
8	7	exlimivv 1889	. 2 ⊢ (∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → 𝐴 ∈ V)
9		eqeq1 2177	. . . . 5 ⊢ (𝑧 = 𝐴 → (𝑧 = ⟨𝑥, 𝑦⟩ ↔ 𝐴 = ⟨𝑥, 𝑦⟩))
10	9	anbi1d 462	. . . 4 ⊢ (𝑧 = 𝐴 → ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ (𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)))
11	10	2exbidv 1861	. . 3 ⊢ (𝑧 = 𝐴 → (∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)))
12		df-opab 4051	. . 3 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
13	11, 12	elab2g 2877	. 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)))
14	1, 8, 13	pm5.21nii 699	1 ⊢ (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))

Syntax hints:   ∧ wa 103   ↔ wb 104   = wceq 1348  ∃wex 1485   ∈ wcel 2141  Vcvv 2730  ⟨cop 3586  {copab 4049

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-opab 4051

This theorem is referenced by:  opelopabsbALT  4244  opelopabsb  4245  opelopabt  4247  opelopabga  4248  opabm  4265  iunopab  4266  epelg  4275  elxp  4628  elco  4777  elcnv  4788  dfmpt3  5320  0neqopab  5898  brabvv  5899  opabex3d  6100  opabex3  6101