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Theorem tpnz 3927
Description: A triplet containing a set is not empty. (Contributed by NM, 10-Apr-1994.)
Hypothesis
Ref Expression
tpnz.1  |-  A  e. 
_V
Assertion
Ref Expression
tpnz  |-  { A ,  B ,  C }  =/=  (/)

Proof of Theorem tpnz
StepHypRef Expression
1 tpnz.1 . . 3  |-  A  e. 
_V
21tpid1 3919 . 2  |-  A  e. 
{ A ,  B ,  C }
3 ne0i 3636 . 2  |-  ( A  e.  { A ,  B ,  C }  ->  { A ,  B ,  C }  =/=  (/) )
42, 3ax-mp 5 1  |-  { A ,  B ,  C }  =/=  (/)
Colors of variables: wff set class
Syntax hints:    e. wcel 1726    =/= wne 2601   _Vcvv 2958   (/)c0 3630   {ctp 3818
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-v 2960  df-dif 3325  df-un 3327  df-nul 3631  df-sn 3822  df-pr 3823  df-tp 3824
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