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Theorem tpnzd 4531
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 4524 . . 3 (𝐴𝑉𝐴 ∈ {𝐵, 𝐶, 𝐴})
3 tprot 4501 . . 3 {𝐴, 𝐵, 𝐶} = {𝐵, 𝐶, 𝐴}
42, 3syl6eleqr 2916 . 2 (𝐴𝑉𝐴 ∈ {𝐴, 𝐵, 𝐶})
5 ne0i 4149 . 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 2166  wne 2998  c0 4143  {ctp 4400
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-ext 2802
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3or 1114  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-clab 2811  df-cleq 2817  df-clel 2820  df-nfc 2957  df-ne 2999  df-v 3415  df-dif 3800  df-un 3802  df-nul 4144  df-sn 4397  df-pr 4399  df-tp 4401
This theorem is referenced by:  raltpd  4532  fr3nr  7239  limsupequzlem  40748  etransclem48  41292
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