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

Theorem prexg 4029
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 3547, prprc1 3545, and prprc2 3546. (Contributed by Jim Kingdon, 16-Sep-2018.)
Assertion
Ref Expression
prexg  |-  ( ( A  e.  V  /\  B  e.  W )  ->  { A ,  B }  e.  _V )

Proof of Theorem prexg
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 preq2 3515 . . . . . 6  |-  ( y  =  B  ->  { x ,  y }  =  { x ,  B } )
21eleq1d 2156 . . . . 5  |-  ( y  =  B  ->  ( { x ,  y }  e.  _V  <->  { x ,  B }  e.  _V ) )
3 zfpair2 4028 . . . . 5  |-  { x ,  y }  e.  _V
42, 3vtoclg 2679 . . . 4  |-  ( B  e.  W  ->  { x ,  B }  e.  _V )
5 preq1 3514 . . . . 5  |-  ( x  =  A  ->  { x ,  B }  =  { A ,  B }
)
65eleq1d 2156 . . . 4  |-  ( x  =  A  ->  ( { x ,  B }  e.  _V  <->  { A ,  B }  e.  _V ) )
74, 6syl5ib 152 . . 3  |-  ( x  =  A  ->  ( B  e.  W  ->  { A ,  B }  e.  _V ) )
87vtocleg 2690 . 2  |-  ( A  e.  V  ->  ( B  e.  W  ->  { A ,  B }  e.  _V ) )
98imp 122 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  { A ,  B }  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1289    e. wcel 1438   _Vcvv 2619   {cpr 3442
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pr 4027
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-un 3001  df-sn 3447  df-pr 3448
This theorem is referenced by:  prelpwi  4032  opexg  4046  opi2  4051  opth  4055  opeqsn  4070  opeqpr  4071  uniop  4073  unex  4257  tpexg  4260  op1stb  4290  op1stbg  4291  onun2  4297  opthreg  4362  relop  4574  acexmidlemv  5632  pr2ne  6799  xrex  9274
  Copyright terms: Public domain W3C validator