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

Theorem dfop 3539
Description: Value of an ordered pair when the arguments are sets, with the conclusion corresponding to Kuratowski's original definition. (Contributed by NM, 25-Jun-1998.)
Hypotheses
Ref Expression
dfop.1 A V
dfop.2 B V
Assertion
Ref Expression
dfop A, B⟩ = {{A}, {A, B}}

Proof of Theorem dfop
StepHypRef Expression
1 dfop.1 . 2 A V
2 dfop.2 . 2 B V
3 dfopg 3538 . 2 ((A V B V) → ⟨A, B⟩ = {{A}, {A, B}})
41, 2, 3mp2an 402 1 A, B⟩ = {{A}, {A, B}}
Colors of variables: wff set class
Syntax hints:   = wceq 1242   wcel 1390  Vcvv 2551  {csn 3367  {cpr 3368  cop 3370
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-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-11 1394  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-v 2553  df-op 3376
This theorem is referenced by:  opid  3558  elop  3959  opi1  3960  opi2  3961  opeqsn  3980  opeqpr  3981  uniop  3983  op1stb  4175  xpsspw  4393  relop  4429  funopg  4877
  Copyright terms: Public domain W3C validator