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Theorem tpnz 4499
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 4491 . 2 𝐴 ∈ {𝐴, 𝐵, 𝐶}
32ne0ii 4122 1 {𝐴, 𝐵, 𝐶} ≠ ∅
Colors of variables: wff setvar class
Syntax hints:  wcel 2158  wne 2977  Vcvv 3390  c0 4113  {ctp 4371
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1880  ax-4 1897  ax-5 2004  ax-6 2070  ax-7 2106  ax-9 2167  ax-10 2187  ax-11 2203  ax-12 2216  ax-13 2422  ax-ext 2784
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-tru 1641  df-ex 1860  df-nf 1865  df-sb 2063  df-clab 2792  df-cleq 2798  df-clel 2801  df-nfc 2936  df-ne 2978  df-v 3392  df-dif 3769  df-un 3771  df-nul 4114  df-sn 4368  df-pr 4370  df-tp 4372
This theorem is referenced by: (None)
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