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Theorem tpss 4728
Description: An unordered triple 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 ((𝐴𝐷𝐵𝐷𝐶𝐷) ↔ {𝐴, 𝐵, 𝐶} ⊆ 𝐷)

Proof of Theorem tpss
StepHypRef Expression
1 unss 4111 . 2 (({𝐴, 𝐵} ⊆ 𝐷 ∧ {𝐶} ⊆ 𝐷) ↔ ({𝐴, 𝐵} ∪ {𝐶}) ⊆ 𝐷)
2 df-3an 1086 . . 3 ((𝐴𝐷𝐵𝐷𝐶𝐷) ↔ ((𝐴𝐷𝐵𝐷) ∧ 𝐶𝐷))
3 tpss.1 . . . . 5 𝐴 ∈ V
4 tpss.2 . . . . 5 𝐵 ∈ V
53, 4prss 4713 . . . 4 ((𝐴𝐷𝐵𝐷) ↔ {𝐴, 𝐵} ⊆ 𝐷)
6 tpss.3 . . . . 5 𝐶 ∈ V
76snss 4679 . . . 4 (𝐶𝐷 ↔ {𝐶} ⊆ 𝐷)
85, 7anbi12i 629 . . 3 (((𝐴𝐷𝐵𝐷) ∧ 𝐶𝐷) ↔ ({𝐴, 𝐵} ⊆ 𝐷 ∧ {𝐶} ⊆ 𝐷))
92, 8bitri 278 . 2 ((𝐴𝐷𝐵𝐷𝐶𝐷) ↔ ({𝐴, 𝐵} ⊆ 𝐷 ∧ {𝐶} ⊆ 𝐷))
10 df-tp 4530 . . 3 {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶})
1110sseq1i 3943 . 2 ({𝐴, 𝐵, 𝐶} ⊆ 𝐷 ↔ ({𝐴, 𝐵} ∪ {𝐶}) ⊆ 𝐷)
121, 9, 113bitr4i 306 1 ((𝐴𝐷𝐵𝐷𝐶𝐷) ↔ {𝐴, 𝐵, 𝐶} ⊆ 𝐷)
Colors of variables: wff setvar class
Syntax hints:  wb 209  wa 399  w3a 1084  wcel 2111  Vcvv 3441  cun 3879  wss 3881  {csn 4525  {cpr 4527  {ctp 4529
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-12 2175  ax-ext 2770
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-v 3443  df-un 3886  df-in 3888  df-ss 3898  df-sn 4526  df-pr 4528  df-tp 4530
This theorem is referenced by:  1cubr  25428  konigsberglem4  28040  rabren3dioph  39751  fourierdlem102  42845  fourierdlem114  42857  nnsum4primesodd  44309  nnsum4primesoddALTV  44310
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