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Theorem tpss 3499
 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 3090 . 2 (({A, B} ⊆ 𝐷 {𝐶} ⊆ 𝐷) ↔ ({A, B} ∪ {𝐶}) ⊆ 𝐷)
2 df-3an 873 . . 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 3490 . . . 4 ((A 𝐷 B 𝐷) ↔ {A, B} ⊆ 𝐷)
6 tpss.3 . . . . 5 𝐶 V
76snss 3464 . . . 4 (𝐶 𝐷 ↔ {𝐶} ⊆ 𝐷)
85, 7anbi12i 436 . . 3 (((A 𝐷 B 𝐷) 𝐶 𝐷) ↔ ({A, B} ⊆ 𝐷 {𝐶} ⊆ 𝐷))
92, 8bitri 173 . 2 ((A 𝐷 B 𝐷 𝐶 𝐷) ↔ ({A, B} ⊆ 𝐷 {𝐶} ⊆ 𝐷))
10 df-tp 3354 . . 3 {A, B, 𝐶} = ({A, B} ∪ {𝐶})
1110sseq1i 2942 . 2 ({A, B, 𝐶} ⊆ 𝐷 ↔ ({A, B} ∪ {𝐶}) ⊆ 𝐷)
121, 9, 113bitr4i 201 1 ((A 𝐷 B 𝐷 𝐶 𝐷) ↔ {A, B, 𝐶} ⊆ 𝐷)
 Colors of variables: wff set class Syntax hints:   ∧ wa 97   ↔ wb 98   ∧ w3a 871   ∈ wcel 1370  Vcvv 2531   ∪ cun 2888   ⊆ wss 2890  {csn 3346  {cpr 3347  {ctp 3348 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000 This theorem depends on definitions:  df-bi 110  df-3an 873  df-tru 1229  df-nf 1326  df-sb 1624  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-v 2533  df-un 2895  df-in 2897  df-ss 2904  df-sn 3352  df-pr 3353  df-tp 3354 This theorem is referenced by: (None)
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