ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  tpss GIF version

Theorem tpss 3651
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 3216 . 2 (({𝐴, 𝐵} ⊆ 𝐷 ∧ {𝐶} ⊆ 𝐷) ↔ ({𝐴, 𝐵} ∪ {𝐶}) ⊆ 𝐷)
2 df-3an 947 . . 3 ((𝐴𝐷𝐵𝐷𝐶𝐷) ↔ ((𝐴𝐷𝐵𝐷) ∧ 𝐶𝐷))
3 tpss.1 . . . . 5 𝐴 ∈ V
4 tpss.2 . . . . 5 𝐵 ∈ V
53, 4prss 3642 . . . 4 ((𝐴𝐷𝐵𝐷) ↔ {𝐴, 𝐵} ⊆ 𝐷)
6 tpss.3 . . . . 5 𝐶 ∈ V
76snss 3615 . . . 4 (𝐶𝐷 ↔ {𝐶} ⊆ 𝐷)
85, 7anbi12i 453 . . 3 (((𝐴𝐷𝐵𝐷) ∧ 𝐶𝐷) ↔ ({𝐴, 𝐵} ⊆ 𝐷 ∧ {𝐶} ⊆ 𝐷))
92, 8bitri 183 . 2 ((𝐴𝐷𝐵𝐷𝐶𝐷) ↔ ({𝐴, 𝐵} ⊆ 𝐷 ∧ {𝐶} ⊆ 𝐷))
10 df-tp 3501 . . 3 {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶})
1110sseq1i 3089 . 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 945  wcel 1463  Vcvv 2657  cun 3035  wss 3037  {csn 3493  {cpr 3494  {ctp 3495
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-v 2659  df-un 3041  df-in 3043  df-ss 3050  df-sn 3499  df-pr 3500  df-tp 3501
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator