Theorem elopab 2110

Description: Membership in a class abstraction of pairs.

Assertion

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

Distinct variable group(s):   x,y,A

Proof of Theorem elopab

Step	Hyp	Ref	Expression
1		elisset 1354	. 2 ⊢ (A ∈ {⟨x, y⟩∣φ} → A ∈ V)
2		opex 1893	. . . . 5 ⊢ ⟨x, y⟩ ∈ V
3		eleq1 1149	. . . . 5 ⊢ (A = ⟨x, y⟩ → (A ∈ V ↔ ⟨x, y⟩ ∈ V))
4	2, 3	mpbiri 169	. . . 4 ⊢ (A = ⟨x, y⟩ → A ∈ V)
5	4	adantr 306	. . 3 ⊢ ((A = ⟨x, y⟩ ∧ φ) → A ∈ V)
6	5	19.23aivv 953	. 2 ⊢ (∃x∃y(A = ⟨x, y⟩ ∧ φ) → A ∈ V)
7		eleq1 1149	. . 3 ⊢ (z = A → (z ∈ {⟨x, y⟩∣φ} ↔ A ∈ {⟨x, y⟩∣φ}))
8		cleq1 1107	. . . . 5 ⊢ (z = A → (z = ⟨x, y⟩ ↔ A = ⟨x, y⟩))
9	8	anbi1d 469	. . . 4 ⊢ (z = A → ((z = ⟨x, y⟩ ∧ φ) ↔ (A = ⟨x, y⟩ ∧ φ)))
10	9	bi2exdv 938	. . 3 ⊢ (z = A → (∃x∃y(z = ⟨x, y⟩ ∧ φ) ↔ ∃x∃y(A = ⟨x, y⟩ ∧ φ)))
11		df-opab 2098	. . . 4 ⊢ {⟨x, y⟩∣φ} = {z∣∃x∃y(z = ⟨x, y⟩ ∧ φ)}
12	11	cleqabi 1176	. . 3 ⊢ (z ∈ {⟨x, y⟩∣φ} ↔ ∃x∃y(z = ⟨x, y⟩ ∧ φ))
13	7, 10, 12	vtoclbg 1384	. 2 ⊢ (A ∈ V → (A ∈ {⟨x, y⟩∣φ} ↔ ∃x∃y(A = ⟨x, y⟩ ∧ φ)))
14	1, 6, 13	pm5.21nii 504	1 ⊢ (A ∈ {⟨x, y⟩∣φ} ↔ ∃x∃y(A = ⟨x, y⟩ ∧ φ))

Syntax hints:   ↔ wb 127   ∧ wa 196  ∃wex 678   = wceq 1091   ∈ wcel 1092  Vcvv 1348  ⟨cop 1810  {copab 2055

This theorem is referenced by:  hbopab 2111  elxp 2442  elcnv 2514

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-opab 2098

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