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Theorem tpssi 4523
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 4341 . 2 {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶})
2 prssi 4508 . . . 4 ((𝐴𝐷𝐵𝐷) → {𝐴, 𝐵} ⊆ 𝐷)
323adant3 1162 . . 3 ((𝐴𝐷𝐵𝐷𝐶𝐷) → {𝐴, 𝐵} ⊆ 𝐷)
4 snssi 4495 . . . 4 (𝐶𝐷 → {𝐶} ⊆ 𝐷)
543ad2ant3 1165 . . 3 ((𝐴𝐷𝐵𝐷𝐶𝐷) → {𝐶} ⊆ 𝐷)
63, 5unssd 3953 . 2 ((𝐴𝐷𝐵𝐷𝐶𝐷) → ({𝐴, 𝐵} ∪ {𝐶}) ⊆ 𝐷)
71, 6syl5eqss 3811 1 ((𝐴𝐷𝐵𝐷𝐶𝐷) → {𝐴, 𝐵, 𝐶} ⊆ 𝐷)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1107  wcel 2155  cun 3732  wss 3734  {csn 4336  {cpr 4338  {ctp 4340
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-ext 2743
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-v 3352  df-un 3739  df-in 3741  df-ss 3748  df-sn 4337  df-pr 4339  df-tp 4341
This theorem is referenced by:  lcmftp  15646  trgcgrg  25717  sgnclre  31072  signstf  31115  limsupequzlem  40618  fourierdlem46  41032  fourierdlem102  41088  fourierdlem114  41100  etransclem48  41162
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