ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  prsstp12 GIF version

Theorem prsstp12 3544
Description: A pair is a subset of an unordered triple containing its members. (Contributed by Jim Kingdon, 11-Aug-2018.)
Assertion
Ref Expression
prsstp12 {𝐴, 𝐵} ⊆ {𝐴, 𝐵, 𝐶}

Proof of Theorem prsstp12
StepHypRef Expression
1 ssun1 3133 . 2 {𝐴, 𝐵} ⊆ ({𝐴, 𝐵} ∪ {𝐶})
2 df-tp 3410 . 2 {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶})
31, 2sseqtr4i 3005 1 {𝐴, 𝐵} ⊆ {𝐴, 𝐵, 𝐶}
Colors of variables: wff set class
Syntax hints:  cun 2942  wss 2944  {csn 3402  {cpr 3403  {ctp 3404
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-un 2949  df-in 2951  df-ss 2958  df-tp 3410
This theorem is referenced by:  prsstp13  3545  prsstp23  3546  sstpr  3555
  Copyright terms: Public domain W3C validator