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Theorem tpnz 3658
 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 3644 . 2 𝐴 ∈ {𝐴, 𝐵, 𝐶}
3 ne0i 3376 . 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 2112   ≠ wne 2310  Vcvv 2691  ∅c0 3370  {ctp 3536 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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-ext 2123 This theorem depends on definitions:  df-bi 116  df-3or 964  df-tru 1335  df-nf 1438  df-sb 1732  df-clab 2128  df-cleq 2134  df-clel 2137  df-nfc 2272  df-ne 2311  df-v 2693  df-dif 3080  df-un 3082  df-nul 3371  df-sn 3540  df-pr 3541  df-tp 3542 This theorem is referenced by: (None)
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