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

Theorem snnz 3610
Description: The singleton of a set is not empty. (Contributed by NM, 10-Apr-1994.)
Hypothesis
Ref Expression
snnz.1  |-  A  e. 
_V
Assertion
Ref Expression
snnz  |-  { A }  =/=  (/)

Proof of Theorem snnz
StepHypRef Expression
1 snnz.1 . 2  |-  A  e. 
_V
2 snnzg 3608 . 2  |-  ( A  e.  _V  ->  { A }  =/=  (/) )
31, 2ax-mp 5 1  |-  { A }  =/=  (/)
Colors of variables: wff set class
Syntax hints:    e. wcel 1463    =/= wne 2283   _Vcvv 2658   (/)c0 3331   {csn 3495
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 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-v 2660  df-dif 3041  df-nul 3332  df-sn 3501
This theorem is referenced by:  0nep0  4057  1n0  6295  ssfii  6828
  Copyright terms: Public domain W3C validator