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

Theorem dfop 3576
 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 3575 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ⟨𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}})
41, 2, 3mp2an 410 1 𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}}
 Colors of variables: wff set class Syntax hints:   = wceq 1259   ∈ wcel 1409  Vcvv 2574  {csn 3403  {cpr 3404  ⟨cop 3406 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-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-11 1413  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-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-v 2576  df-op 3412 This theorem is referenced by:  opid  3595  elop  3996  opi1  3997  opi2  3998  opeqsn  4017  opeqpr  4018  uniop  4020  op1stb  4237  xpsspw  4478  relop  4514  funopg  4962
 Copyright terms: Public domain W3C validator