Theorem elopab 4696

Description: Membership in a class abstraction of 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 2867	. 2 ⊢ (A ∈ {⟨x, y⟩ ∣ φ} → A ∈ V)
2		vex 2862	. . . . . 6 ⊢ x ∈ V
3		vex 2862	. . . . . 6 ⊢ y ∈ V
4	2, 3	opex 4588	. . . . 5 ⊢ ⟨x, y⟩ ∈ V
5		eleq1 2413	. . . . 5 ⊢ (A = ⟨x, y⟩ → (A ∈ V ↔ ⟨x, y⟩ ∈ V))
6	4, 5	mpbiri 224	. . . 4 ⊢ (A = ⟨x, y⟩ → A ∈ V)
7	6	adantr 451	. . 3 ⊢ ((A = ⟨x, y⟩ ∧ φ) → A ∈ V)
8	7	exlimivv 1635	. 2 ⊢ (∃x∃y(A = ⟨x, y⟩ ∧ φ) → A ∈ V)
9		eqeq1 2359	. . . . 5 ⊢ (z = A → (z = ⟨x, y⟩ ↔ A = ⟨x, y⟩))
10	9	anbi1d 685	. . . 4 ⊢ (z = A → ((z = ⟨x, y⟩ ∧ φ) ↔ (A = ⟨x, y⟩ ∧ φ)))
11	10	2exbidv 1628	. . 3 ⊢ (z = A → (∃x∃y(z = ⟨x, y⟩ ∧ φ) ↔ ∃x∃y(A = ⟨x, y⟩ ∧ φ)))
12		df-opab 4623	. . 3 ⊢ {⟨x, y⟩ ∣ φ} = {z ∣ ∃x∃y(z = ⟨x, y⟩ ∧ φ)}
13	11, 12	elab2g 2987	. 2 ⊢ (A ∈ V → (A ∈ {⟨x, y⟩ ∣ φ} ↔ ∃x∃y(A = ⟨x, y⟩ ∧ φ)))
14	1, 8, 13	pm5.21nii 342	1 ⊢ (A ∈ {⟨x, y⟩ ∣ φ} ↔ ∃x∃y(A = ⟨x, y⟩ ∧ φ))

Syntax hints:   ↔ wb 176   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  Vcvv 2859  ⟨cop 4561  {copab 4622

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-addc 4378  df-nnc 4379  df-phi 4565  df-op 4566  df-opab 4623

This theorem is referenced by:  opelopabt  4699  opelopabga  4700  opabn0  4716  el1st  4729  setconslem4  4734  elswap  4740  elxp  4801  elcnv  4889  composeex  5820