Theorem prexg 4296

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 3777, prprc1 3775, and prprc2 3776. (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 3744	. . . . . 6 ⊢ (𝑦 = 𝐵 → {𝑥, 𝑦} = {𝑥, 𝐵})
2	1	eleq1d 2298	. . . . 5 ⊢ (𝑦 = 𝐵 → ({𝑥, 𝑦} ∈ V ↔ {𝑥, 𝐵} ∈ V))
3		zfpair2 4295	. . . . 5 ⊢ {𝑥, 𝑦} ∈ V
4	2, 3	vtoclg 2861	. . . 4 ⊢ (𝐵 ∈ 𝑊 → {𝑥, 𝐵} ∈ V)
5		preq1 3743	. . . . 5 ⊢ (𝑥 = 𝐴 → {𝑥, 𝐵} = {𝐴, 𝐵})
6	5	eleq1d 2298	. . . 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 2874	. 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 1395   ∈ wcel 2200  Vcvv 2799  {cpr 3667

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pr 4294

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-un 3201  df-sn 3672  df-pr 3673

This theorem is referenced by:  prelpw  4300  prelpwi  4301  opexg  4315  opi2  4320  opth  4324  opeqsn  4340  opeqpr  4341  uniop  4343  unex  4533  tpexg  4536  op1stb  4570  op1stbg  4571  onun2  4583  opthreg  4649  relop  4875  acexmidlemv  6008  2oex  6590  en2prd  6983  pw2f1odclem  7008  pr2ne  7381  exmidonfinlem  7387  exmidaclem  7406  sup3exmid  9120  xrex  10069  2strbasg  13174  2stropg  13175  prdsex  13323  prdsval  13327  xpsfval  13402  xpsval  13406  gsumprval  13453  struct2slots2dom  15860  structiedg0val  15862  edgstruct  15885  umgrbien  15931  upgr1edc  15942  upgr1eopdc  15944  uspgr1edc  16059  usgr1e  16060  uspgr1eopdc  16062  uspgr1ewopdc  16063  usgr1eop  16064  usgr2v1e2w  16065  isomninnlem  16512  trilpolemlt1  16523  iswomninnlem  16531  iswomni0  16533  ismkvnnlem  16534