MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  tpssi Structured version   Visualization version   GIF version

Theorem tpssi 4838
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 4631 . 2 {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶})
2 prssi 4821 . . . 4 ((𝐴𝐷𝐵𝐷) → {𝐴, 𝐵} ⊆ 𝐷)
323adant3 1133 . . 3 ((𝐴𝐷𝐵𝐷𝐶𝐷) → {𝐴, 𝐵} ⊆ 𝐷)
4 snssi 4808 . . . 4 (𝐶𝐷 → {𝐶} ⊆ 𝐷)
543ad2ant3 1136 . . 3 ((𝐴𝐷𝐵𝐷𝐶𝐷) → {𝐶} ⊆ 𝐷)
63, 5unssd 4192 . 2 ((𝐴𝐷𝐵𝐷𝐶𝐷) → ({𝐴, 𝐵} ∪ {𝐶}) ⊆ 𝐷)
71, 6eqsstrid 4022 1 ((𝐴𝐷𝐵𝐷𝐶𝐷) → {𝐴, 𝐵, 𝐶} ⊆ 𝐷)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1087  wcel 2108  cun 3949  wss 3951  {csn 4626  {cpr 4628  {ctp 4630
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-v 3482  df-un 3956  df-ss 3968  df-sn 4627  df-pr 4629  df-tp 4631
This theorem is referenced by:  lcmftp  16673  trgcgrg  28523  cyc3co2  33160  sgnclre  34542  signstf  34581  limsupequzlem  45737  fourierdlem46  46167  fourierdlem102  46223  fourierdlem114  46235  etransclem48  46297  grtrissvtx  47911  grtrimap  47915  usgrexmpl2nb0  47990  usgrexmpl2nb3  47993
  Copyright terms: Public domain W3C validator