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Theorem prexgOLD 3946
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 3480, prprc1 3478, and prprc2 3479. This is a special case of prexg 3947 and new proofs should use prexg 3947 instead. (Contributed by Jim Kingdon, 25-Jul-2019.) (New usage is discouraged.) (Proof modification is discouraged.) TODO: replace its uses by uses of prexg 3947 and then remove it.
Assertion
Ref Expression
prexgOLD ((𝐴 ∈ V ∧ 𝐵 ∈ V) → {𝐴, 𝐵} ∈ V)

Proof of Theorem prexgOLD
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 preq2 3448 . . . . . 6 (𝑦 = 𝐵 → {𝑥, 𝑦} = {𝑥, 𝐵})
21eleq1d 2106 . . . . 5 (𝑦 = 𝐵 → ({𝑥, 𝑦} ∈ V ↔ {𝑥, 𝐵} ∈ V))
3 zfpair2 3945 . . . . 5 {𝑥, 𝑦} ∈ V
42, 3vtoclg 2613 . . . 4 (𝐵 ∈ V → {𝑥, 𝐵} ∈ V)
5 preq1 3447 . . . . 5 (𝑥 = 𝐴 → {𝑥, 𝐵} = {𝐴, 𝐵})
65eleq1d 2106 . . . 4 (𝑥 = 𝐴 → ({𝑥, 𝐵} ∈ V ↔ {𝐴, 𝐵} ∈ V))
74, 6syl5ib 143 . . 3 (𝑥 = 𝐴 → (𝐵 ∈ V → {𝐴, 𝐵} ∈ V))
87vtocleg 2624 . 2 (𝐴 ∈ V → (𝐵 ∈ V → {𝐴, 𝐵} ∈ V))
98imp 115 1 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → {𝐴, 𝐵} ∈ V)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 97   = wceq 1243  wcel 1393  Vcvv 2557  {cpr 3376
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pr 3944
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-un 2922  df-sn 3381  df-pr 3382
This theorem is referenced by:  prelpwi  3950  opexgOLD  3965  opi2  3970  opth  3974  opeqsn  3989  opeqpr  3990  uniop  3992  unex  4176  op1stb  4209  op1stbg  4210  opthreg  4280  relop  4486
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