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Theorem ssopab2 4003
Description: Equivalence of ordered pair abstraction subclass and implication. (Contributed by NM, 27-Dec-1996.) (Revised by Mario Carneiro, 19-May-2013.)
Assertion
Ref Expression
ssopab2  { <. ,  >.  |  }  C_  { <. ,  >.  |  }

Proof of Theorem ssopab2
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 nfa1 1431 . . . 4  F/
2 nfa1 1431 . . . . . 6  F/
3 sp 1398 . . . . . . 7
43anim2d 320 . . . . . 6  <. , 
>.  <. ,  >.
52, 4eximd 1500 . . . . 5  <. ,  >.  <. ,  >.
65sps 1427 . . . 4 
<. ,  >.  <. ,  >.
71, 6eximd 1500 . . 3  <. ,  >.  <. ,  >.
87ss2abdv 3007 . 2  {  | 
<. ,  >.  }  C_  {  |  <. ,  >.  }
9 df-opab 3810 . 2  { <. ,  >.  |  }  {  |  <. ,  >.  }
10 df-opab 3810 . 2  { <. ,  >.  |  }  {  |  <. ,  >.  }
118, 9, 103sstr4g 2980 1  { <. ,  >.  |  }  C_  { <. ,  >.  |  }
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97  wal 1240   wceq 1242  wex 1378   {cab 2023    C_ wss 2911   <.cop 3370   {copab 3808
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  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-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-in 2918  df-ss 2925  df-opab 3810
This theorem is referenced by:  ssopab2b  4004  ssopab2i  4005  ssopab2dv  4006
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