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Theorem tpnz 3837
Description: A triplet containing a set is not empty. (Contributed by NM, 10-Apr-1994.)
Hypothesis
Ref Expression
tpnz.1 A V
Assertion
Ref Expression
tpnz {A, B, C} ≠

Proof of Theorem tpnz
StepHypRef Expression
1 tpnz.1 . . 3 A V
21tpid1 3829 . 2 A {A, B, C}
3 ne0i 3556 . 2 (A {A, B, C} → {A, B, C} ≠ )
42, 3ax-mp 5 1 {A, B, C} ≠
Colors of variables: wff setvar class
Syntax hints:   wcel 1710  wne 2516  Vcvv 2859  c0 3550  {ctp 3739
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-nul 3551  df-sn 3741  df-pr 3742  df-tp 3743
This theorem is referenced by: (None)
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