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

Theorem tpssi 4768
 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 4569 . 2 {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶})
2 prssi 4753 . . . 4 ((𝐴𝐷𝐵𝐷) → {𝐴, 𝐵} ⊆ 𝐷)
323adant3 1126 . . 3 ((𝐴𝐷𝐵𝐷𝐶𝐷) → {𝐴, 𝐵} ⊆ 𝐷)
4 snssi 4740 . . . 4 (𝐶𝐷 → {𝐶} ⊆ 𝐷)
543ad2ant3 1129 . . 3 ((𝐴𝐷𝐵𝐷𝐶𝐷) → {𝐶} ⊆ 𝐷)
63, 5unssd 4166 . 2 ((𝐴𝐷𝐵𝐷𝐶𝐷) → ({𝐴, 𝐵} ∪ {𝐶}) ⊆ 𝐷)
71, 6eqsstrid 4019 1 ((𝐴𝐷𝐵𝐷𝐶𝐷) → {𝐴, 𝐵, 𝐶} ⊆ 𝐷)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ w3a 1081   ∈ wcel 2107   ∪ cun 3938   ⊆ wss 3940  {csn 4564  {cpr 4566  {ctp 4568 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-v 3502  df-un 3945  df-in 3947  df-ss 3956  df-sn 4565  df-pr 4567  df-tp 4569 This theorem is referenced by:  lcmftp  15970  trgcgrg  26218  cyc3co2  30699  sgnclre  31686  signstf  31725  limsupequzlem  41871  fourierdlem46  42306  fourierdlem102  42362  fourierdlem114  42374  etransclem48  42436
 Copyright terms: Public domain W3C validator