Theorem prexg 4260

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 3745, prprc1 3743, and prprc2 3744. (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.

Step	Hyp	Ref	Expression
1		preq2 3713	. . . . . 6 ⊢ (𝑦 = 𝐵 → {𝑥, 𝑦} = {𝑥, 𝐵})
2	1	eleq1d 2275	. . . . 5 ⊢ (𝑦 = 𝐵 → ({𝑥, 𝑦} ∈ V ↔ {𝑥, 𝐵} ∈ V))
3		zfpair2 4259	. . . . 5 ⊢ {𝑥, 𝑦} ∈ V
4	2, 3	vtoclg 2835	. . . 4 ⊢ (𝐵 ∈ 𝑊 → {𝑥, 𝐵} ∈ V)
5		preq1 3712	. . . . 5 ⊢ (𝑥 = 𝐴 → {𝑥, 𝐵} = {𝐴, 𝐵})
6	5	eleq1d 2275	. . . 4 ⊢ (𝑥 = 𝐴 → ({𝑥, 𝐵} ∈ V ↔ {𝐴, 𝐵} ∈ V))
7	4, 6	imbitrid 154	. . 3 ⊢ (𝑥 = 𝐴 → (𝐵 ∈ 𝑊 → {𝐴, 𝐵} ∈ V))
8	7	vtocleg 2846	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐵 ∈ 𝑊 → {𝐴, 𝐵} ∈ V))
9	8	imp 124	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → {𝐴, 𝐵} ∈ V)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1373   ∈ wcel 2177  Vcvv 2773  {cpr 3636

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2180  ax-ext 2188  ax-sep 4167  ax-pr 4258

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-v 2775  df-un 3172  df-sn 3641  df-pr 3642

This theorem is referenced by:  prelpwi  4263  opexg  4277  opi2  4282  opth  4286  opeqsn  4302  opeqpr  4303  uniop  4305  unex  4493  tpexg  4496  op1stb  4530  op1stbg  4531  onun2  4543  opthreg  4609  relop  4833  acexmidlemv  5952  en2prd  6920  pw2f1odclem  6943  pr2ne  7312  exmidonfinlem  7314  exmidaclem  7333  sup3exmid  9043  xrex  9991  2strbasg  13002  2stropg  13003  prdsex  13151  prdsval  13155  xpsfval  13230  xpsval  13234  gsumprval  13281  struct2slots2dom  15687  structiedg0val  15689  edgstruct  15710  umgrbien  15756  upgr1edc  15764  upgr1eopdc  15766  isomninnlem  16084  trilpolemlt1  16095  iswomninnlem  16103  iswomni0  16105  ismkvnnlem  16106