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Theorem dford3 26533
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 4406 . . 3  |-  ( Ord 
N  ->  Tr  N
)
2 ordelord 4414 . . . . 5  |-  ( ( Ord  N  /\  x  e.  N )  ->  Ord  x )
3 ordtr 4406 . . . . 5  |-  ( Ord  x  ->  Tr  x
)
42, 3syl 15 . . . 4  |-  ( ( Ord  N  /\  x  e.  N )  ->  Tr  x )
54ralrimiva 2626 . . 3  |-  ( Ord 
N  ->  A. x  e.  N  Tr  x
)
61, 5jca 518 . 2  |-  ( Ord 
N  ->  ( Tr  N  /\  A. x  e.  N  Tr  x ) )
7 simpl 443 . . 3  |-  ( ( Tr  N  /\  A. x  e.  N  Tr  x )  ->  Tr  N )
8 dford3lem1 26531 . . . . 5  |-  ( ( Tr  N  /\  A. x  e.  N  Tr  x )  ->  A. a  e.  N  ( Tr  a  /\  A. x  e.  a  Tr  x ) )
9 dford3lem2 26532 . . . . . 6  |-  ( ( Tr  a  /\  A. x  e.  a  Tr  x )  ->  a  e.  On )
109ralimi 2618 . . . . 5  |-  ( A. a  e.  N  ( Tr  a  /\  A. x  e.  a  Tr  x
)  ->  A. a  e.  N  a  e.  On )
118, 10syl 15 . . . 4  |-  ( ( Tr  N  /\  A. x  e.  N  Tr  x )  ->  A. a  e.  N  a  e.  On )
12 dfss3 3170 . . . 4  |-  ( N 
C_  On  <->  A. a  e.  N  a  e.  On )
1311, 12sylibr 203 . . 3  |-  ( ( Tr  N  /\  A. x  e.  N  Tr  x )  ->  N  C_  On )
14 ordon 4574 . . . 4  |-  Ord  On
1514a1i 10 . . 3  |-  ( ( Tr  N  /\  A. x  e.  N  Tr  x )  ->  Ord  On )
16 trssord 4409 . . 3  |-  ( ( Tr  N  /\  N  C_  On  /\  Ord  On )  ->  Ord  N )
177, 13, 15, 16syl3anc 1182 . 2  |-  ( ( Tr  N  /\  A. x  e.  N  Tr  x )  ->  Ord  N )
186, 17impbii 180 1  |-  ( Ord 
N  <->  ( Tr  N  /\  A. x  e.  N  Tr  x ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    e. wcel 1684   A.wral 2543    C_ wss 3152   Tr wtr 4113   Ord word 4391   Oncon0 4392
This theorem is referenced by:  dford4  26534
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-un 4512  ax-reg 7306
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-tr 4114  df-eprel 4305  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-suc 4398
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