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

Theorem elpri 3454
Description: If a class is an element of a pair, then it is one of the two paired elements. (Contributed by Scott Fenton, 1-Apr-2011.)
Assertion
Ref Expression
elpri  |-  ( A  e.  { B ,  C }  ->  ( A  =  B  \/  A  =  C ) )

Proof of Theorem elpri
StepHypRef Expression
1 elprg 3451 . 2  |-  ( A  e.  { B ,  C }  ->  ( A  e.  { B ,  C }  <->  ( A  =  B  \/  A  =  C ) ) )
21ibi 174 1  |-  ( A  e.  { B ,  C }  ->  ( A  =  B  \/  A  =  C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 662    = wceq 1287    e. wcel 1436   {cpr 3432
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-v 2617  df-un 2992  df-sn 3437  df-pr 3438
This theorem is referenced by:  nelpri  3455  opth1  4037  0nelop  4049  ontr2exmid  4314  onintexmid  4361  reg3exmidlemwe  4367  funtpg  5030  ftpg  5444  acexmidlemcase  5608  2oconcl  6157  en2eqpr  6575  eldju1st  6706  finomni  6740  exmidomniim  6741  exmidfodomrlemr  6772  exmidfodomrlemrALT  6773  m1expcl2  9875  maxleim  10533  maxleast  10541  minmax  10554  el2oss1o  11325  nninfalllem1  11337  nninfall  11338  nninfsellemqall  11345  nninfomnilem  11348
  Copyright terms: Public domain W3C validator