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Theorem tpnzd 4784
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 4773 . 2 (𝐴𝑉𝐴 ∈ {𝐴, 𝐵, 𝐶})
3 ne0i 4334 . 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 2105  wne 2939  c0 4322  {ctp 4632
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-ne 2940  df-v 3475  df-dif 3951  df-un 3953  df-nul 4323  df-sn 4629  df-pr 4631  df-tp 4633
This theorem is referenced by:  raltpd  4785  fr3nr  7763  limsupequzlem  44897  etransclem48  45457
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