Theorem prexg 4301

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 3782, prprc1 3780, and prprc2 3781. (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 3749	. . . . . 6 ⊢ (𝑦 = 𝐵 → {𝑥, 𝑦} = {𝑥, 𝐵})
2	1	eleq1d 2300	. . . . 5 ⊢ (𝑦 = 𝐵 → ({𝑥, 𝑦} ∈ V ↔ {𝑥, 𝐵} ∈ V))
3		zfpair2 4300	. . . . 5 ⊢ {𝑥, 𝑦} ∈ V
4	2, 3	vtoclg 2864	. . . 4 ⊢ (𝐵 ∈ 𝑊 → {𝑥, 𝐵} ∈ V)
5		preq1 3748	. . . . 5 ⊢ (𝑥 = 𝐴 → {𝑥, 𝐵} = {𝐴, 𝐵})
6	5	eleq1d 2300	. . . 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 2877	. 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 1397   ∈ wcel 2202  Vcvv 2802  {cpr 3670

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-un 3204  df-sn 3675  df-pr 3676

This theorem is referenced by:  prelpw  4305  prelpwi  4306  opexg  4320  opi2  4325  opth  4329  opeqsn  4345  opeqpr  4346  uniop  4348  unex  4538  tpexg  4541  op1stb  4575  op1stbg  4576  onun2  4588  opthreg  4654  relop  4880  acexmidlemv  6016  2oex  6599  en2prd  6992  pw2f1odclem  7020  pr2ne  7397  exmidonfinlem  7404  exmidaclem  7423  sup3exmid  9137  xrex  10091  2strbasg  13208  2stropg  13209  prdsex  13357  prdsval  13361  xpsfval  13436  xpsval  13440  gsumprval  13487  struct2slots2dom  15895  structiedg0val  15897  edgstruct  15921  umgrbien  15967  upgr1edc  15978  upgr1eopdc  15980  uspgr1edc  16097  usgr1e  16098  uspgr1eopdc  16100  uspgr1ewopdc  16101  usgr1eop  16102  usgr2v1e2w  16103  vdegp1aid  16171  vdegp1bid  16172  eupth2lemsfi  16335  isomninnlem  16660  trilpolemlt1  16671  iswomninnlem  16679  iswomni0  16681  ismkvnnlem  16682