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Theorem tpnz 3893
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 3885 . 2  |-  A  e. 
{ A ,  B ,  C }
3 ne0i 3602 . 2  |-  ( A  e.  { A ,  B ,  C }  ->  { A ,  B ,  C }  =/=  (/) )
42, 3ax-mp 8 1  |-  { A ,  B ,  C }  =/=  (/)
Colors of variables: wff set class
Syntax hints:    e. wcel 1721    =/= wne 2575   _Vcvv 2924   (/)c0 3596   {ctp 3784
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-v 2926  df-dif 3291  df-un 3293  df-nul 3597  df-sn 3788  df-pr 3789  df-tp 3790
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