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

Theorem tpss 3520
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 3111 . 2 (({A, B} ⊆ 𝐷 {𝐶} ⊆ 𝐷) ↔ ({A, B} ∪ {𝐶}) ⊆ 𝐷)
2 df-3an 886 . . 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 3511 . . . 4 ((A 𝐷 B 𝐷) ↔ {A, B} ⊆ 𝐷)
6 tpss.3 . . . . 5 𝐶 V
76snss 3485 . . . 4 (𝐶 𝐷 ↔ {𝐶} ⊆ 𝐷)
85, 7anbi12i 433 . . 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 3375 . . 3 {A, B, 𝐶} = ({A, B} ∪ {𝐶})
1110sseq1i 2963 . 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 884   wcel 1390  Vcvv 2551  cun 2909  wss 2911  {csn 3367  {cpr 3368  {ctp 3369
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-sn 3373  df-pr 3374  df-tp 3375
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator