Theorem elopab 4697

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 2868	. 2 ⊢ (A ∈ {⟨x, y⟩ ∣ φ} → A ∈ V)
2		vex 2863	. . . . . 6 ⊢ x ∈ V
3		vex 2863	. . . . . 6 ⊢ y ∈ V
4	2, 3	opex 4589	. . . . 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 4624	. . 3 ⊢ {⟨x, y⟩ ∣ φ} = {z ∣ ∃x∃y(z = ⟨x, y⟩ ∧ φ)}
13	11, 12	elab2g 2988	. 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 2860  ⟨cop 4562  {copab 4623

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 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

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 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-addc 4379  df-nnc 4380  df-phi 4566  df-op 4567  df-opab 4624

This theorem is referenced by:  opelopabt  4700  opelopabga  4701  opabn0  4717  el1st  4730  setconslem4  4735  elswap  4741  elxp  4802  elcnv  4890  composeex  5821