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Theorem prexg 4211
Description: The Axiom of Pairing using class variables. Theorem 7.13 of [Quine] p. 51, but restricted to classes which exist. For proper classes, see prprc 3702, prprc1 3700, and prprc2 3701. (Contributed by Jim Kingdon, 16-Sep-2018.)
Assertion
Ref Expression
prexg ((𝐴𝑉𝐵𝑊) → {𝐴, 𝐵} ∈ V)

Proof of Theorem prexg
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 preq2 3670 . . . . . 6 (𝑦 = 𝐵 → {𝑥, 𝑦} = {𝑥, 𝐵})
21eleq1d 2246 . . . . 5 (𝑦 = 𝐵 → ({𝑥, 𝑦} ∈ V ↔ {𝑥, 𝐵} ∈ V))
3 zfpair2 4210 . . . . 5 {𝑥, 𝑦} ∈ V
42, 3vtoclg 2797 . . . 4 (𝐵𝑊 → {𝑥, 𝐵} ∈ V)
5 preq1 3669 . . . . 5 (𝑥 = 𝐴 → {𝑥, 𝐵} = {𝐴, 𝐵})
65eleq1d 2246 . . . 4 (𝑥 = 𝐴 → ({𝑥, 𝐵} ∈ V ↔ {𝐴, 𝐵} ∈ V))
74, 6imbitrid 154 . . 3 (𝑥 = 𝐴 → (𝐵𝑊 → {𝐴, 𝐵} ∈ V))
87vtocleg 2808 . 2 (𝐴𝑉 → (𝐵𝑊 → {𝐴, 𝐵} ∈ V))
98imp 124 1 ((𝐴𝑉𝐵𝑊) → {𝐴, 𝐵} ∈ V)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104   = wceq 1353  wcel 2148  Vcvv 2737  {cpr 3593
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pr 4209
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739  df-un 3133  df-sn 3598  df-pr 3599
This theorem is referenced by:  prelpwi  4214  opexg  4228  opi2  4233  opth  4237  opeqsn  4252  opeqpr  4253  uniop  4255  unex  4441  tpexg  4444  op1stb  4478  op1stbg  4479  onun2  4489  opthreg  4555  relop  4777  acexmidlemv  5872  pr2ne  7190  exmidonfinlem  7191  exmidaclem  7206  sup3exmid  8913  xrex  9855  2strbasg  12577  2stropg  12578  prdsex  12717  xpsfval  12766  xpsval  12770  isomninnlem  14748  trilpolemlt1  14759  iswomninnlem  14767  iswomni0  14769  ismkvnnlem  14770
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