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Theorem tpnz 3729
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 3715 . 2 𝐴 ∈ {𝐴, 𝐵, 𝐶}
3 ne0i 3441 . 2 (𝐴 ∈ {𝐴, 𝐵, 𝐶} → {𝐴, 𝐵, 𝐶} ≠ ∅)
42, 3ax-mp 5 1 {𝐴, 𝐵, 𝐶} ≠ ∅
Colors of variables: wff set class
Syntax hints:  wcel 2158  wne 2357  Vcvv 2749  c0 3434  {ctp 3606
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169
This theorem depends on definitions:  df-bi 117  df-3or 980  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-v 2751  df-dif 3143  df-un 3145  df-nul 3435  df-sn 3610  df-pr 3611  df-tp 3612
This theorem is referenced by: (None)
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