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Theorem tpss 4831
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 4177 . 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 4816 . . . 4 ((𝐴𝐷𝐵𝐷) ↔ {𝐴, 𝐵} ⊆ 𝐷)
6 tpss.3 . . . . 5 𝐶 ∈ V
76snss 4782 . . . 4 (𝐶𝐷 ↔ {𝐶} ⊆ 𝐷)
85, 7anbi12i 626 . . 3 (((𝐴𝐷𝐵𝐷) ∧ 𝐶𝐷) ↔ ({𝐴, 𝐵} ⊆ 𝐷 ∧ {𝐶} ⊆ 𝐷))
92, 8bitri 275 . 2 ((𝐴𝐷𝐵𝐷𝐶𝐷) ↔ ({𝐴, 𝐵} ⊆ 𝐷 ∧ {𝐶} ⊆ 𝐷))
10 df-tp 4626 . . 3 {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶})
1110sseq1i 4003 . 2 ({𝐴, 𝐵, 𝐶} ⊆ 𝐷 ↔ ({𝐴, 𝐵} ∪ {𝐶}) ⊆ 𝐷)
121, 9, 113bitr4i 303 1 ((𝐴𝐷𝐵𝐷𝐶𝐷) ↔ {𝐴, 𝐵, 𝐶} ⊆ 𝐷)
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 395  w3a 1084  wcel 2098  Vcvv 3466  cun 3939  wss 3941  {csn 4621  {cpr 4623  {ctp 4625
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-v 3468  df-un 3946  df-in 3948  df-ss 3958  df-sn 4622  df-pr 4624  df-tp 4626
This theorem is referenced by:  1cubr  26693  konigsberglem4  29980  rabren3dioph  42067  fourierdlem102  45434  fourierdlem114  45446  nnsum4primesodd  46974  nnsum4primesoddALTV  46975
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