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Theorem prexg 3974
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 3507, prprc1 3505, and prprc2 3506. (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 3475 . . . . . 6 (𝑦 = 𝐵 → {𝑥, 𝑦} = {𝑥, 𝐵})
21eleq1d 2122 . . . . 5 (𝑦 = 𝐵 → ({𝑥, 𝑦} ∈ V ↔ {𝑥, 𝐵} ∈ V))
3 zfpair2 3972 . . . . 5 {𝑥, 𝑦} ∈ V
42, 3vtoclg 2630 . . . 4 (𝐵𝑊 → {𝑥, 𝐵} ∈ V)
5 preq1 3474 . . . . 5 (𝑥 = 𝐴 → {𝑥, 𝐵} = {𝐴, 𝐵})
65eleq1d 2122 . . . 4 (𝑥 = 𝐴 → ({𝑥, 𝐵} ∈ V ↔ {𝐴, 𝐵} ∈ V))
74, 6syl5ib 147 . . 3 (𝑥 = 𝐴 → (𝐵𝑊 → {𝐴, 𝐵} ∈ V))
87vtocleg 2641 . 2 (𝐴𝑉 → (𝐵𝑊 → {𝐴, 𝐵} ∈ V))
98imp 119 1 ((𝐴𝑉𝐵𝑊) → {𝐴, 𝐵} ∈ V)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101   = wceq 1259  wcel 1409  Vcvv 2574  {cpr 3403
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-pr 3971
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-un 2949  df-sn 3408  df-pr 3409
This theorem is referenced by:  opexg  3991  tpexg  4206  onun2  4243  acexmidlemv  5537  xrex  8856
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