MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  prnzg Unicode version

Theorem prnzg 3746
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 3706 . . 3  |-  ( x  =  A  ->  { x ,  B }  =  { A ,  B }
)
21neeq1d 2459 . 2  |-  ( x  =  A  ->  ( { x ,  B }  =/=  (/)  <->  { A ,  B }  =/=  (/) ) )
3 vex 2791 . . 3  |-  x  e. 
_V
43prnz 3745 . 2  |-  { x ,  B }  =/=  (/)
52, 4vtoclg 2843 1  |-  ( A  e.  V  ->  { A ,  B }  =/=  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684    =/= wne 2446   (/)c0 3455   {cpr 3641
This theorem is referenced by:  0nelop  4256  fr2nr  4371  mreincl  13501  subrgin  15568  lssincl  15722  incld  16780  difelsiga  23494  umgra1  23878  inttop4  25517  inidl  26655  uslgra1  28118  usgra1  28119  pmapmeet  29962  diameetN  31246  dihmeetlem2N  31489  dihmeetcN  31492  dihmeet  31533
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-v 2790  df-dif 3155  df-un 3157  df-nul 3456  df-sn 3646  df-pr 3647
  Copyright terms: Public domain W3C validator