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Theorem tpss 4520
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 ((𝐴𝐷𝐵𝐷𝐶𝐷) ↔ {𝐴, 𝐵, 𝐶} ⊆ 𝐷)

Proof of Theorem tpss
StepHypRef Expression
1 unss 3949 . 2 (({𝐴, 𝐵} ⊆ 𝐷 ∧ {𝐶} ⊆ 𝐷) ↔ ({𝐴, 𝐵} ∪ {𝐶}) ⊆ 𝐷)
2 df-3an 1109 . . 3 ((𝐴𝐷𝐵𝐷𝐶𝐷) ↔ ((𝐴𝐷𝐵𝐷) ∧ 𝐶𝐷))
3 tpss.1 . . . . 5 𝐴 ∈ V
4 tpss.2 . . . . 5 𝐵 ∈ V
53, 4prss 4505 . . . 4 ((𝐴𝐷𝐵𝐷) ↔ {𝐴, 𝐵} ⊆ 𝐷)
6 tpss.3 . . . . 5 𝐶 ∈ V
76snss 4470 . . . 4 (𝐶𝐷 ↔ {𝐶} ⊆ 𝐷)
85, 7anbi12i 620 . . 3 (((𝐴𝐷𝐵𝐷) ∧ 𝐶𝐷) ↔ ({𝐴, 𝐵} ⊆ 𝐷 ∧ {𝐶} ⊆ 𝐷))
92, 8bitri 266 . 2 ((𝐴𝐷𝐵𝐷𝐶𝐷) ↔ ({𝐴, 𝐵} ⊆ 𝐷 ∧ {𝐶} ⊆ 𝐷))
10 df-tp 4339 . . 3 {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶})
1110sseq1i 3789 . 2 ({𝐴, 𝐵, 𝐶} ⊆ 𝐷 ↔ ({𝐴, 𝐵} ∪ {𝐶}) ⊆ 𝐷)
121, 9, 113bitr4i 294 1 ((𝐴𝐷𝐵𝐷𝐶𝐷) ↔ {𝐴, 𝐵, 𝐶} ⊆ 𝐷)
Colors of variables: wff setvar class
Syntax hints:  wb 197  wa 384  w3a 1107  wcel 2155  Vcvv 3350  cun 3730  wss 3732  {csn 4334  {cpr 4336  {ctp 4338
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-ext 2743
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-v 3352  df-un 3737  df-in 3739  df-ss 3746  df-sn 4335  df-pr 4337  df-tp 4339
This theorem is referenced by:  1cubr  24860  konigsberglem4  27533  rabren3dioph  38057  fourierdlem102  41062  fourierdlem114  41074  nnsum4primesodd  42360  nnsum4primesoddALTV  42361
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