New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  tpss GIF version

Theorem tpss 3871
 Description: A triplet of elements of a class is a subset of the class. (Contributed by NM, 9-Apr-1994.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
Hypotheses
Ref Expression
tpss.1 A V
tpss.2 B V
tpss.3 C V
Assertion
Ref Expression
tpss ((A D B D C D) ↔ {A, B, C} D)

Proof of Theorem tpss
StepHypRef Expression
1 unss 3437 . 2 (({A, B} D {C} D) ↔ ({A, B} ∪ {C}) D)
2 df-3an 936 . . 3 ((A D B D C D) ↔ ((A D B D) C D))
3 tpss.1 . . . . 5 A V
4 tpss.2 . . . . 5 B V
53, 4prss 3861 . . . 4 ((A D B D) ↔ {A, B} D)
6 tpss.3 . . . . 5 C V
76snss 3838 . . . 4 (C D ↔ {C} D)
85, 7anbi12i 678 . . 3 (((A D B D) C D) ↔ ({A, B} D {C} D))
92, 8bitri 240 . 2 ((A D B D C D) ↔ ({A, B} D {C} D))
10 df-tp 3743 . . 3 {A, B, C} = ({A, B} ∪ {C})
1110sseq1i 3295 . 2 ({A, B, C} D ↔ ({A, B} ∪ {C}) D)
121, 9, 113bitr4i 268 1 ((A D B D C D) ↔ {A, B, C} D)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176   ∧ wa 358   ∧ w3a 934   ∈ wcel 1710  Vcvv 2859   ∪ cun 3207   ⊆ wss 3257  {csn 3737  {cpr 3738  {ctp 3739 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-ss 3259  df-sn 3741  df-pr 3742  df-tp 3743 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator