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Theorem tpss 3694
 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 3256 . 2 (({𝐴, 𝐵} ⊆ 𝐷 ∧ {𝐶} ⊆ 𝐷) ↔ ({𝐴, 𝐵} ∪ {𝐶}) ⊆ 𝐷)
2 df-3an 965 . . 3 ((𝐴𝐷𝐵𝐷𝐶𝐷) ↔ ((𝐴𝐷𝐵𝐷) ∧ 𝐶𝐷))
3 tpss.1 . . . . 5 𝐴 ∈ V
4 tpss.2 . . . . 5 𝐵 ∈ V
53, 4prss 3685 . . . 4 ((𝐴𝐷𝐵𝐷) ↔ {𝐴, 𝐵} ⊆ 𝐷)
6 tpss.3 . . . . 5 𝐶 ∈ V
76snss 3658 . . . 4 (𝐶𝐷 ↔ {𝐶} ⊆ 𝐷)
85, 7anbi12i 456 . . 3 (((𝐴𝐷𝐵𝐷) ∧ 𝐶𝐷) ↔ ({𝐴, 𝐵} ⊆ 𝐷 ∧ {𝐶} ⊆ 𝐷))
92, 8bitri 183 . 2 ((𝐴𝐷𝐵𝐷𝐶𝐷) ↔ ({𝐴, 𝐵} ⊆ 𝐷 ∧ {𝐶} ⊆ 𝐷))
10 df-tp 3541 . . 3 {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶})
1110sseq1i 3129 . 2 ({𝐴, 𝐵, 𝐶} ⊆ 𝐷 ↔ ({𝐴, 𝐵} ∪ {𝐶}) ⊆ 𝐷)
121, 9, 113bitr4i 211 1 ((𝐴𝐷𝐵𝐷𝐶𝐷) ↔ {𝐴, 𝐵, 𝐶} ⊆ 𝐷)
 Colors of variables: wff set class Syntax hints:   ∧ wa 103   ↔ wb 104   ∧ w3a 963   ∈ wcel 1481  Vcvv 2690   ∪ cun 3075   ⊆ wss 3077  {csn 3533  {cpr 3534  {ctp 3535 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2692  df-un 3081  df-in 3083  df-ss 3090  df-sn 3539  df-pr 3540  df-tp 3541 This theorem is referenced by: (None)
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