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Theorem tpnzd 4783
Description: An unordered triple containing a set is not empty. (Contributed by Thierry Arnoux, 8-Apr-2019.)
Hypothesis
Ref Expression
tpnzd.1 (𝜑𝐴𝑉)
Assertion
Ref Expression
tpnzd (𝜑 → {𝐴, 𝐵, 𝐶} ≠ ∅)

Proof of Theorem tpnzd
StepHypRef Expression
1 tpnzd.1 . 2 (𝜑𝐴𝑉)
2 tpid1g 4772 . 2 (𝐴𝑉𝐴 ∈ {𝐴, 𝐵, 𝐶})
3 ne0i 4333 . 2 (𝐴 ∈ {𝐴, 𝐵, 𝐶} → {𝐴, 𝐵, 𝐶} ≠ ∅)
41, 2, 33syl 18 1 (𝜑 → {𝐴, 𝐵, 𝐶} ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2107  wne 2941  c0 4321  {ctp 4631
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-v 3477  df-dif 3950  df-un 3952  df-nul 4322  df-sn 4628  df-pr 4630  df-tp 4632
This theorem is referenced by:  raltpd  4784  fr3nr  7754  limsupequzlem  44373  etransclem48  44933
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