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

Theorem prnzg 3617
Description: A pair containing a set is not empty. (Contributed by FL, 19-Sep-2011.)
Assertion
Ref Expression
prnzg  |-  ( A  e.  V  ->  { A ,  B }  =/=  (/) )

Proof of Theorem prnzg
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 preq1 3570 . . 3  |-  ( x  =  A  ->  { x ,  B }  =  { A ,  B }
)
21neeq1d 2303 . 2  |-  ( x  =  A  ->  ( { x ,  B }  =/=  (/)  <->  { A ,  B }  =/=  (/) ) )
3 vex 2663 . . 3  |-  x  e. 
_V
43prnz 3615 . 2  |-  { x ,  B }  =/=  (/)
52, 4vtoclg 2720 1  |-  ( A  e.  V  ->  { A ,  B }  =/=  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1316    e. wcel 1465    =/= wne 2285   (/)c0 3333   {cpr 3498
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-v 2662  df-dif 3043  df-un 3045  df-nul 3334  df-sn 3503  df-pr 3504
This theorem is referenced by:  0nelop  4140
  Copyright terms: Public domain W3C validator