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Theorem opeqexb 4620
 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 ↔ xy A = x, y)
Distinct variable group:   x,A,y

Proof of Theorem opeqexb
StepHypRef Expression
1 opexb 4603 . 2 ( Proj1 A, Proj2 A V ↔ ( Proj1 A V Proj2 A V))
2 opeq 4619 . . 3 A = Proj1 A, Proj2 A
32eleq1i 2416 . 2 (A V ↔ Proj1 A, Proj2 A V)
4 eeanv 1913 . . 3 (xy(x = Proj1 A y = Proj2 A) ↔ (x x = Proj1 A y y = Proj2 A))
52eqeq1i 2360 . . . . 5 (A = x, y Proj1 A, Proj2 A = x, y)
6 eqcom 2355 . . . . 5 ( Proj1 A, Proj2 A = x, yx, y = Proj1 A, Proj2 A)
7 opth 4602 . . . . 5 (x, y = Proj1 A, Proj2 A ↔ (x = Proj1 A y = Proj2 A))
85, 6, 73bitri 262 . . . 4 (A = x, y ↔ (x = Proj1 A y = Proj2 A))
982exbii 1583 . . 3 (xy A = x, yxy(x = Proj1 A y = Proj2 A))
10 isset 2863 . . . 4 ( Proj1 A V ↔ x x = Proj1 A)
11 isset 2863 . . . 4 ( Proj2 A V ↔ y y = Proj2 A)
1210, 11anbi12i 678 . . 3 (( Proj1 A V Proj2 A V) ↔ (x x = Proj1 A y y = Proj2 A))
134, 9, 123bitr4i 268 . 2 (xy A = x, y ↔ ( Proj1 A V Proj2 A V))
141, 3, 133bitr4i 268 1 (A V ↔ xy A = x, y)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  Vcvv 2859  ⟨cop 4561   Proj1 cproj1 4563   Proj2 cproj2 4564 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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 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-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 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  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-pss 3261  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-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568 This theorem is referenced by:  opeqex  4621  eliunxp  4821  dmsnn0  5064
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