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

Theorem prprc 3714
Description: An unordered pair containing two proper classes is the empty set. (Contributed by NM, 22-Mar-2006.)
Assertion
Ref Expression
prprc  |-  ( ( -.  A  e.  _V  /\ 
-.  B  e.  _V )  ->  { A ,  B }  =  (/) )

Proof of Theorem prprc
StepHypRef Expression
1 prprc1 3712 . 2  |-  ( -.  A  e.  _V  ->  { A ,  B }  =  { B } )
2 snprc 3669 . . 3  |-  ( -.  B  e.  _V  <->  { B }  =  (/) )
32biimpi 120 . 2  |-  ( -.  B  e.  _V  ->  { B }  =  (/) )
41, 3sylan9eq 2240 1  |-  ( ( -.  A  e.  _V  /\ 
-.  B  e.  _V )  ->  { A ,  B }  =  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 104    = wceq 1363    e. wcel 2158   _Vcvv 2749   (/)c0 3434   {csn 3604   {cpr 3605
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-in1 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169
This theorem depends on definitions:  df-bi 117  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-v 2751  df-dif 3143  df-un 3145  df-nul 3435  df-sn 3610  df-pr 3611
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator