ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  preq1 Unicode version

Theorem preq1 3502
Description: Equality theorem for unordered pairs. (Contributed by NM, 29-Mar-1998.)
Assertion
Ref Expression
preq1  |-  ( A  =  B  ->  { A ,  C }  =  { B ,  C }
)

Proof of Theorem preq1
StepHypRef Expression
1 sneq 3442 . . 3  |-  ( A  =  B  ->  { A }  =  { B } )
21uneq1d 3142 . 2  |-  ( A  =  B  ->  ( { A }  u.  { C } )  =  ( { B }  u.  { C } ) )
3 df-pr 3438 . 2  |-  { A ,  C }  =  ( { A }  u.  { C } )
4 df-pr 3438 . 2  |-  { B ,  C }  =  ( { B }  u.  { C } )
52, 3, 43eqtr4g 2142 1  |-  ( A  =  B  ->  { A ,  C }  =  { B ,  C }
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1287    u. cun 2986   {csn 3431   {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:  preq2  3503  preq12  3504  preq1i  3505  preq1d  3508  tpeq1  3511  prnzg  3547  preq12b  3597  preq12bg  3600  opeq1  3605  uniprg  3651  intprg  3704  prexg  4012  opthreg  4345  bj-prexg  11240
  Copyright terms: Public domain W3C validator