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

Theorem prssi 3550
Description: A pair of elements of a class is a subset of the class. (Contributed by NM, 16-Jan-2015.)
Assertion
Ref Expression
prssi ((𝐴𝐶𝐵𝐶) → {𝐴, 𝐵} ⊆ 𝐶)

Proof of Theorem prssi
StepHypRef Expression
1 prssg 3549 . 2 ((𝐴𝐶𝐵𝐶) → ((𝐴𝐶𝐵𝐶) ↔ {𝐴, 𝐵} ⊆ 𝐶))
21ibi 169 1 ((𝐴𝐶𝐵𝐶) → {𝐴, 𝐵} ⊆ 𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wcel 1409  wss 2945  {cpr 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 2950  df-in 2952  df-ss 2959  df-sn 3409  df-pr 3410
This theorem is referenced by:  tpssi  3558  prelpwi  3978  onun2  4244  onintexmid  4325  nnregexmid  4370  m1expcl2  9442  m1expcl  9443  bdop  10382
  Copyright terms: Public domain W3C validator