Theorem opeqexb 4621

Description: A class is a set iff it is equal to an ordered pair. (Contributed by Scott Fenton, 19-Apr-2021.)

Assertion

Ref	Expression
opeqexb	⊢ (A ∈ V ↔ ∃x∃y A = ⟨x, y⟩)

Distinct variable group:   x,A,y

Proof of Theorem opeqexb

Step	Hyp	Ref	Expression
1		opexb 4604	. 2 ⊢ (⟨ Proj1 A, Proj2 A⟩ ∈ V ↔ ( Proj1 A ∈ V ∧ Proj2 A ∈ V))
2		opeq 4620	. . 3 ⊢ A = ⟨ Proj1 A, Proj2 A⟩
3	2	eleq1i 2416	. 2 ⊢ (A ∈ V ↔ ⟨ Proj1 A, Proj2 A⟩ ∈ V)
4		eeanv 1913	. . 3 ⊢ (∃x∃y(x = Proj1 A ∧ y = Proj2 A) ↔ (∃x x = Proj1 A ∧ ∃y y = Proj2 A))
5	2	eqeq1i 2360	. . . . 5 ⊢ (A = ⟨x, y⟩ ↔ ⟨ Proj1 A, Proj2 A⟩ = ⟨x, y⟩)
6		eqcom 2355	. . . . 5 ⊢ (⟨ Proj1 A, Proj2 A⟩ = ⟨x, y⟩ ↔ ⟨x, y⟩ = ⟨ Proj1 A, Proj2 A⟩)
7		opth 4603	. . . . 5 ⊢ (⟨x, y⟩ = ⟨ Proj1 A, Proj2 A⟩ ↔ (x = Proj1 A ∧ y = Proj2 A))
8	5, 6, 7	3bitri 262	. . . 4 ⊢ (A = ⟨x, y⟩ ↔ (x = Proj1 A ∧ y = Proj2 A))
9	8	2exbii 1583	. . 3 ⊢ (∃x∃y A = ⟨x, y⟩ ↔ ∃x∃y(x = Proj1 A ∧ y = Proj2 A))
10		isset 2864	. . . 4 ⊢ ( Proj1 A ∈ V ↔ ∃x x = Proj1 A)
11		isset 2864	. . . 4 ⊢ ( Proj2 A ∈ V ↔ ∃y y = Proj2 A)
12	10, 11	anbi12i 678	. . 3 ⊢ (( Proj1 A ∈ V ∧ Proj2 A ∈ V) ↔ (∃x x = Proj1 A ∧ ∃y y = Proj2 A))
13	4, 9, 12	3bitr4i 268	. 2 ⊢ (∃x∃y A = ⟨x, y⟩ ↔ ( Proj1 A ∈ V ∧ Proj2 A ∈ V))
14	1, 3, 13	3bitr4i 268	1 ⊢ (A ∈ V ↔ ∃x∃y A = ⟨x, y⟩)

Syntax hints:   ↔ wb 176   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  Vcvv 2860  ⟨cop 4562   Proj1 cproj1 4564   Proj2 cproj2 4565

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-13 1712  ax-14 1714  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-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  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-pss 3262  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-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569

This theorem is referenced by:  opeqex  4622  eliunxp  4822  dmsnn0  5065