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Theorem oprcl 3567
Description: If an ordered pair has an element, then its arguments are sets. (Contributed by Mario Carneiro, 26-Apr-2015.)
Assertion
Ref Expression
oprcl  C  <. ,  >.  _V  _V

Proof of Theorem oprcl
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex2 2567 . 2  C  <. ,  >.  <. ,  >.
2 df-op 3379 . . . . . . 7  <. ,  >.  {  |  _V  _V  { { } ,  { ,  } } }
32eleq2i 2104 . . . . . 6  <. ,  >.  {  |  _V  _V  { { } ,  { ,  } } }
4 df-clab 2027 . . . . . 6  {  |  _V  _V  { { } ,  { ,  } } }  _V  _V  { { } ,  { ,  } }
53, 4bitri 173 . . . . 5  <. ,  >.  _V  _V  { { } ,  { ,  } }
6 3simpa 901 . . . . . 6  _V  _V  { { } ,  { ,  } }  _V  _V
76sbimi 1647 . . . . 5  _V  _V 
{ { } ,  { ,  } }  _V  _V
85, 7sylbi 114 . . . 4  <. ,  >.  _V  _V
9 nfv 1421 . . . . 5  F/  _V  _V
109sbf 1660 . . . 4  _V  _V  _V  _V
118, 10sylib 127 . . 3  <. ,  >.  _V  _V
1211exlimiv 1489 . 2  <. ,  >.  _V  _V
131, 12syl 14 1  C  <. ,  >.  _V  _V
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   w3a 885  wex 1381   wcel 1393  wsb 1645   {cab 2026   _Vcvv 2554   {csn 3370   {cpr 3371   <.cop 3373
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 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-3an 887  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-v 2556  df-op 3379
This theorem is referenced by:  opth1  3967  opth  3968  0nelop  3979
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