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Theorem tpnzd 4257
 Description: A triplet 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 tpid3g 4248 . . 3 (𝐴𝑉𝐴 ∈ {𝐵, 𝐶, 𝐴})
3 tprot 4228 . . 3 {𝐴, 𝐵, 𝐶} = {𝐵, 𝐶, 𝐴}
42, 3syl6eleqr 2699 . 2 (𝐴𝑉𝐴 ∈ {𝐴, 𝐵, 𝐶})
5 ne0i 3880 . 2 (𝐴 ∈ {𝐴, 𝐵, 𝐶} → {𝐴, 𝐵, 𝐶} ≠ ∅)
61, 4, 53syl 18 1 (𝜑 → {𝐴, 𝐵, 𝐶} ≠ ∅)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 1977   ≠ wne 2780  ∅c0 3874  {ctp 4129 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-v 3175  df-dif 3543  df-un 3545  df-nul 3875  df-sn 4126  df-pr 4128  df-tp 4130 This theorem is referenced by:  raltpd  4258  fr3nr  6871  etransclem48  39175
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