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Theorem opid 3594
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 3416 . . . 4 {𝐴} = {𝐴, 𝐴}
21eqcomi 2060 . . 3 {𝐴, 𝐴} = {𝐴}
32preq2i 3478 . 2 {{𝐴}, {𝐴, 𝐴}} = {{𝐴}, {𝐴}}
4 opid.1 . . 3 𝐴 ∈ V
54, 4dfop 3575 . 2 𝐴, 𝐴⟩ = {{𝐴}, {𝐴, 𝐴}}
6 dfsn2 3416 . 2 {{𝐴}} = {{𝐴}, {𝐴}}
73, 5, 63eqtr4i 2086 1 𝐴, 𝐴⟩ = {{𝐴}}
Colors of variables: wff set class
Syntax hints:   = wceq 1259  wcel 1409  Vcvv 2574  {csn 3402  {cpr 3403  cop 3405
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-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  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-nfc 2183  df-v 2576  df-un 2949  df-sn 3408  df-pr 3409  df-op 3411
This theorem is referenced by:  dmsnsnsng  4825  funopg  4961
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