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Theorem dford3 26791
Description: Ordinals are precisely the hereditarily transitive classes. (Contributed by Stefan O'Rear, 28-Oct-2014.)
Assertion
Ref Expression
dford3  |-  ( Ord 
N  <->  ( Tr  N  /\  A. x  e.  N  Tr  x ) )
Distinct variable group:    x, N

Proof of Theorem dford3
Dummy variable  a is distinct from all other variables.
StepHypRef Expression
1 ordtr 4537 . . 3  |-  ( Ord 
N  ->  Tr  N
)
2 ordelord 4545 . . . . 5  |-  ( ( Ord  N  /\  x  e.  N )  ->  Ord  x )
3 ordtr 4537 . . . . 5  |-  ( Ord  x  ->  Tr  x
)
42, 3syl 16 . . . 4  |-  ( ( Ord  N  /\  x  e.  N )  ->  Tr  x )
54ralrimiva 2733 . . 3  |-  ( Ord 
N  ->  A. x  e.  N  Tr  x
)
61, 5jca 519 . 2  |-  ( Ord 
N  ->  ( Tr  N  /\  A. x  e.  N  Tr  x ) )
7 simpl 444 . . 3  |-  ( ( Tr  N  /\  A. x  e.  N  Tr  x )  ->  Tr  N )
8 dford3lem1 26789 . . . . 5  |-  ( ( Tr  N  /\  A. x  e.  N  Tr  x )  ->  A. a  e.  N  ( Tr  a  /\  A. x  e.  a  Tr  x ) )
9 dford3lem2 26790 . . . . . 6  |-  ( ( Tr  a  /\  A. x  e.  a  Tr  x )  ->  a  e.  On )
109ralimi 2725 . . . . 5  |-  ( A. a  e.  N  ( Tr  a  /\  A. x  e.  a  Tr  x
)  ->  A. a  e.  N  a  e.  On )
118, 10syl 16 . . . 4  |-  ( ( Tr  N  /\  A. x  e.  N  Tr  x )  ->  A. a  e.  N  a  e.  On )
12 dfss3 3282 . . . 4  |-  ( N 
C_  On  <->  A. a  e.  N  a  e.  On )
1311, 12sylibr 204 . . 3  |-  ( ( Tr  N  /\  A. x  e.  N  Tr  x )  ->  N  C_  On )
14 ordon 4704 . . . 4  |-  Ord  On
1514a1i 11 . . 3  |-  ( ( Tr  N  /\  A. x  e.  N  Tr  x )  ->  Ord  On )
16 trssord 4540 . . 3  |-  ( ( Tr  N  /\  N  C_  On  /\  Ord  On )  ->  Ord  N )
177, 13, 15, 16syl3anc 1184 . 2  |-  ( ( Tr  N  /\  A. x  e.  N  Tr  x )  ->  Ord  N )
186, 17impbii 181 1  |-  ( Ord 
N  <->  ( Tr  N  /\  A. x  e.  N  Tr  x ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    e. wcel 1717   A.wral 2650    C_ wss 3264   Tr wtr 4244   Ord word 4522   Oncon0 4523
This theorem is referenced by:  dford4  26792
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pr 4345  ax-un 4642  ax-reg 7494
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-rab 2659  df-v 2902  df-sbc 3106  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-br 4155  df-opab 4209  df-tr 4245  df-eprel 4436  df-po 4445  df-so 4446  df-fr 4483  df-we 4485  df-ord 4526  df-on 4527  df-suc 4529
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