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Theorem tpssi 4503
 Description: A triple of elements of a class is a subset of the class. (Contributed by Alexander van der Vekens, 1-Feb-2018.)
Assertion
Ref Expression
tpssi ((𝐴𝐷𝐵𝐷𝐶𝐷) → {𝐴, 𝐵, 𝐶} ⊆ 𝐷)

Proof of Theorem tpssi
StepHypRef Expression
1 df-tp 4322 . 2 {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶})
2 prssi 4488 . . . 4 ((𝐴𝐷𝐵𝐷) → {𝐴, 𝐵} ⊆ 𝐷)
323adant3 1126 . . 3 ((𝐴𝐷𝐵𝐷𝐶𝐷) → {𝐴, 𝐵} ⊆ 𝐷)
4 snssi 4475 . . . 4 (𝐶𝐷 → {𝐶} ⊆ 𝐷)
543ad2ant3 1129 . . 3 ((𝐴𝐷𝐵𝐷𝐶𝐷) → {𝐶} ⊆ 𝐷)
63, 5unssd 3940 . 2 ((𝐴𝐷𝐵𝐷𝐶𝐷) → ({𝐴, 𝐵} ∪ {𝐶}) ⊆ 𝐷)
71, 6syl5eqss 3798 1 ((𝐴𝐷𝐵𝐷𝐶𝐷) → {𝐴, 𝐵, 𝐶} ⊆ 𝐷)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ w3a 1071   ∈ wcel 2145   ∪ cun 3721   ⊆ wss 3723  {csn 4317  {cpr 4319  {ctp 4321 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 837  df-3an 1073  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-v 3353  df-un 3728  df-in 3730  df-ss 3737  df-sn 4318  df-pr 4320  df-tp 4322 This theorem is referenced by:  lcmftp  15557  trgcgrg  25631  sgnclre  30941  signstf  30983  limsupequzlem  40469  fourierdlem46  40883  fourierdlem102  40939  fourierdlem114  40951  etransclem48  41013
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