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Theorem tpnzd 4448
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 4441 . . 3 (𝐴𝑉𝐴 ∈ {𝐵, 𝐶, 𝐴})
3 tprot 4420 . . 3 {𝐴, 𝐵, 𝐶} = {𝐵, 𝐶, 𝐴}
42, 3syl6eleqr 2861 . 2 (𝐴𝑉𝐴 ∈ {𝐴, 𝐵, 𝐶})
5 ne0i 4069 . 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 2145  wne 2943  c0 4063  {ctp 4320
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3or 1072  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-v 3353  df-dif 3726  df-un 3728  df-nul 4064  df-sn 4317  df-pr 4319  df-tp 4321
This theorem is referenced by:  raltpd  4449  fr3nr  7126  limsupequzlem  40472  etransclem48  41016
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