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

Theorem prexg 3938
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 3471, prprc1 3469, and prprc2 3470. (Contributed by Jim Kingdon, 16-Sep-2018.)
Assertion
Ref Expression
prexg ((A 𝑉 B 𝑊) → {A, B} V)

Proof of Theorem prexg
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 preq2 3439 . . . . . 6 (y = B → {x, y} = {x, B})
21eleq1d 2103 . . . . 5 (y = B → ({x, y} V ↔ {x, B} V))
3 zfpair2 3936 . . . . 5 {x, y} V
42, 3vtoclg 2607 . . . 4 (B 𝑊 → {x, B} V)
5 preq1 3438 . . . . 5 (x = A → {x, B} = {A, B})
65eleq1d 2103 . . . 4 (x = A → ({x, B} V ↔ {A, B} V))
74, 6syl5ib 143 . . 3 (x = A → (B 𝑊 → {A, B} V))
87vtocleg 2618 . 2 (A 𝑉 → (B 𝑊 → {A, B} V))
98imp 115 1 ((A 𝑉 B 𝑊) → {A, B} V)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1242   wcel 1390  Vcvv 2551  {cpr 3368
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-un 2916  df-sn 3373  df-pr 3374
This theorem is referenced by:  opexg  3955  tpexg  4145  onun2  4182  acexmidlemv  5453  xrex  8526
  Copyright terms: Public domain W3C validator