ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  prexg GIF version

Theorem prexg 4103
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 3603, prprc1 3601, and prprc2 3602. (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 3571 . . . . . 6 (𝑦 = 𝐵 → {𝑥, 𝑦} = {𝑥, 𝐵})
21eleq1d 2186 . . . . 5 (𝑦 = 𝐵 → ({𝑥, 𝑦} ∈ V ↔ {𝑥, 𝐵} ∈ V))
3 zfpair2 4102 . . . . 5 {𝑥, 𝑦} ∈ V
42, 3vtoclg 2720 . . . 4 (𝐵𝑊 → {𝑥, 𝐵} ∈ V)
5 preq1 3570 . . . . 5 (𝑥 = 𝐴 → {𝑥, 𝐵} = {𝐴, 𝐵})
65eleq1d 2186 . . . 4 (𝑥 = 𝐴 → ({𝑥, 𝐵} ∈ V ↔ {𝐴, 𝐵} ∈ V))
74, 6syl5ib 153 . . 3 (𝑥 = 𝐴 → (𝐵𝑊 → {𝐴, 𝐵} ∈ V))
87vtocleg 2731 . 2 (𝐴𝑉 → (𝐵𝑊 → {𝐴, 𝐵} ∈ V))
98imp 123 1 ((𝐴𝑉𝐵𝑊) → {𝐴, 𝐵} ∈ V)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1316  wcel 1465  Vcvv 2660  {cpr 3498
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662  df-un 3045  df-sn 3503  df-pr 3504
This theorem is referenced by:  prelpwi  4106  opexg  4120  opi2  4125  opth  4129  opeqsn  4144  opeqpr  4145  uniop  4147  unex  4332  tpexg  4335  op1stb  4369  op1stbg  4370  onun2  4376  opthreg  4441  relop  4659  acexmidlemv  5740  pr2ne  7016  exmidonfinlem  7017  exmidaclem  7032  sup3exmid  8683  xrex  9607  2strbasg  11987  2stropg  11988  isomninnlem  13152  trilpolemlt1  13161
  Copyright terms: Public domain W3C validator