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

Theorem opid 3567
 Description: The ordered pair ⟨𝐴, 𝐴⟩ in Kuratowski's representation. (Contributed by FL, 28-Dec-2011.)
Hypothesis
Ref Expression
opid.1 𝐴 ∈ V
Assertion
Ref Expression
opid 𝐴, 𝐴⟩ = {{𝐴}}

Proof of Theorem opid
StepHypRef Expression
1 dfsn2 3389 . . . 4 {𝐴} = {𝐴, 𝐴}
21eqcomi 2044 . . 3 {𝐴, 𝐴} = {𝐴}
32preq2i 3451 . 2 {{𝐴}, {𝐴, 𝐴}} = {{𝐴}, {𝐴}}
4 opid.1 . . 3 𝐴 ∈ V
54, 4dfop 3548 . 2 𝐴, 𝐴⟩ = {{𝐴}, {𝐴, 𝐴}}
6 dfsn2 3389 . 2 {{𝐴}} = {{𝐴}, {𝐴}}
73, 5, 63eqtr4i 2070 1 𝐴, 𝐴⟩ = {{𝐴}}
 Colors of variables: wff set class Syntax hints:   = wceq 1243   ∈ wcel 1393  Vcvv 2557  {csn 3375  {cpr 3376  ⟨cop 3378 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-3an 887  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-sn 3381  df-pr 3382  df-op 3384 This theorem is referenced by:  dmsnsnsng  4798  funopg  4934
 Copyright terms: Public domain W3C validator