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Theorem tpnzd 4312
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 4303 . . 3 (𝐴𝑉𝐴 ∈ {𝐵, 𝐶, 𝐴})
3 tprot 4282 . . 3 {𝐴, 𝐵, 𝐶} = {𝐵, 𝐶, 𝐴}
42, 3syl6eleqr 2711 . 2 (𝐴𝑉𝐴 ∈ {𝐴, 𝐵, 𝐶})
5 ne0i 3919 . 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 1989  wne 2793  c0 3913  {ctp 4179
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-v 3200  df-dif 3575  df-un 3577  df-nul 3914  df-sn 4176  df-pr 4178  df-tp 4180
This theorem is referenced by:  raltpd  4313  fr3nr  6976  limsupequzlem  39760  etransclem48  40268
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