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Theorem tpnz 3618
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 3604 . 2  |-  A  e. 
{ A ,  B ,  C }
3 ne0i 3339 . 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 1465    =/= wne 2285   _Vcvv 2660   (/)c0 3333   {ctp 3499
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-3or 948  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  df-tp 3505
This theorem is referenced by: (None)
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