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Theorem prexg 4002
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 3526, prprc1 3524, and prprc2 3525. (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 3494 . . . . . 6 (𝑦 = 𝐵 → {𝑥, 𝑦} = {𝑥, 𝐵})
21eleq1d 2151 . . . . 5 (𝑦 = 𝐵 → ({𝑥, 𝑦} ∈ V ↔ {𝑥, 𝐵} ∈ V))
3 zfpair2 4001 . . . . 5 {𝑥, 𝑦} ∈ V
42, 3vtoclg 2669 . . . 4 (𝐵𝑊 → {𝑥, 𝐵} ∈ V)
5 preq1 3493 . . . . 5 (𝑥 = 𝐴 → {𝑥, 𝐵} = {𝐴, 𝐵})
65eleq1d 2151 . . . 4 (𝑥 = 𝐴 → ({𝑥, 𝐵} ∈ V ↔ {𝐴, 𝐵} ∈ V))
74, 6syl5ib 152 . . 3 (𝑥 = 𝐴 → (𝐵𝑊 → {𝐴, 𝐵} ∈ V))
87vtocleg 2680 . 2 (𝐴𝑉 → (𝐵𝑊 → {𝐴, 𝐵} ∈ V))
98imp 122 1 ((𝐴𝑉𝐵𝑊) → {𝐴, 𝐵} ∈ V)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102   = wceq 1285  wcel 1434  Vcvv 2612  {cpr 3423
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3922  ax-pr 4000
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2614  df-un 2988  df-sn 3428  df-pr 3429
This theorem is referenced by:  prelpwi  4005  opexg  4019  opi2  4024  opth  4028  opeqsn  4043  opeqpr  4044  uniop  4046  unex  4230  tpexg  4233  op1stb  4263  op1stbg  4264  onun2  4270  opthreg  4335  relop  4544  acexmidlemv  5589  pr2ne  6723  xrex  9200
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