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Theorem tpnz 3614
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 3600 . 2  |-  A  e. 
{ A ,  B ,  C }
3 ne0i 3335 . 2  |-  ( A  e.  { A ,  B ,  C }  ->  { A ,  B ,  C }  =/=  (/) )
42, 3ax-mp 7 1  |-  { A ,  B ,  C }  =/=  (/)
Colors of variables: wff set class
Syntax hints:    e. wcel 1463    =/= wne 2282   _Vcvv 2657   (/)c0 3329   {ctp 3495
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 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-3or 946  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ne 2283  df-v 2659  df-dif 3039  df-un 3041  df-nul 3330  df-sn 3499  df-pr 3500  df-tp 3501
This theorem is referenced by: (None)
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