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

Theorem prssi 3522
Description: A pair of elements of a class is a subset of the class. (Contributed by NM, 16-Jan-2015.)
Assertion
Ref Expression
prssi  |-  ( ( A  e.  C  /\  B  e.  C )  ->  { A ,  B }  C_  C )

Proof of Theorem prssi
StepHypRef Expression
1 prssg 3521 . 2  |-  ( ( A  e.  C  /\  B  e.  C )  ->  ( ( A  e.  C  /\  B  e.  C )  <->  { A ,  B }  C_  C
) )
21ibi 165 1  |-  ( ( A  e.  C  /\  B  e.  C )  ->  { A ,  B }  C_  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 97    e. wcel 1393    C_ wss 2917   {cpr 3376
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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-sn 3381  df-pr 3382
This theorem is referenced by:  tpssi  3530  prelpwi  3950  onun2  4216  onintexmid  4297  nnregexmid  4342  m1expcl2  9277  m1expcl  9278  bdop  9995
  Copyright terms: Public domain W3C validator