Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  snsstp3 Structured version   GIF version

Theorem snsstp3 3507
 Description: A singleton is a subset of an unordered triple containing its member. (Contributed by NM, 9-Oct-2013.)
Assertion
Ref Expression
snsstp3 {𝐶} ⊆ {A, B, 𝐶}

Proof of Theorem snsstp3
StepHypRef Expression
1 ssun2 3101 . 2 {𝐶} ⊆ ({A, B} ∪ {𝐶})
2 df-tp 3375 . 2 {A, B, 𝐶} = ({A, B} ∪ {𝐶})
31, 2sseqtr4i 2972 1 {𝐶} ⊆ {A, B, 𝐶}
 Colors of variables: wff set class Syntax hints:   ∪ 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-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-tp 3375 This theorem is referenced by:  sstpr  3519
 Copyright terms: Public domain W3C validator