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Theorem tpssi 4843
Description: An unordered 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 4636 . 2 {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶})
2 prssi 4826 . . . 4 ((𝐴𝐷𝐵𝐷) → {𝐴, 𝐵} ⊆ 𝐷)
323adant3 1131 . . 3 ((𝐴𝐷𝐵𝐷𝐶𝐷) → {𝐴, 𝐵} ⊆ 𝐷)
4 snssi 4813 . . . 4 (𝐶𝐷 → {𝐶} ⊆ 𝐷)
543ad2ant3 1134 . . 3 ((𝐴𝐷𝐵𝐷𝐶𝐷) → {𝐶} ⊆ 𝐷)
63, 5unssd 4202 . 2 ((𝐴𝐷𝐵𝐷𝐶𝐷) → ({𝐴, 𝐵} ∪ {𝐶}) ⊆ 𝐷)
71, 6eqsstrid 4044 1 ((𝐴𝐷𝐵𝐷𝐶𝐷) → {𝐴, 𝐵, 𝐶} ⊆ 𝐷)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1086  wcel 2106  cun 3961  wss 3963  {csn 4631  {cpr 4633  {ctp 4635
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-v 3480  df-un 3968  df-ss 3980  df-sn 4632  df-pr 4634  df-tp 4636
This theorem is referenced by:  lcmftp  16670  trgcgrg  28538  cyc3co2  33143  sgnclre  34521  signstf  34560  limsupequzlem  45678  fourierdlem46  46108  fourierdlem102  46164  fourierdlem114  46176  etransclem48  46238  grtrissvtx  47849  grtrimap  47851  usgrexmpl2nb0  47926  usgrexmpl2nb3  47929
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