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Theorem dford3 27211
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 4630 . . 3  |-  ( Ord 
N  ->  Tr  N
)
2 ordelord 4638 . . . . 5  |-  ( ( Ord  N  /\  x  e.  N )  ->  Ord  x )
3 ordtr 4630 . . . . 5  |-  ( Ord  x  ->  Tr  x
)
42, 3syl 16 . . . 4  |-  ( ( Ord  N  /\  x  e.  N )  ->  Tr  x )
54ralrimiva 2796 . . 3  |-  ( Ord 
N  ->  A. x  e.  N  Tr  x
)
61, 5jca 520 . 2  |-  ( Ord 
N  ->  ( Tr  N  /\  A. x  e.  N  Tr  x ) )
7 simpl 445 . . 3  |-  ( ( Tr  N  /\  A. x  e.  N  Tr  x )  ->  Tr  N )
8 dford3lem1 27209 . . . . 5  |-  ( ( Tr  N  /\  A. x  e.  N  Tr  x )  ->  A. a  e.  N  ( Tr  a  /\  A. x  e.  a  Tr  x ) )
9 dford3lem2 27210 . . . . . 6  |-  ( ( Tr  a  /\  A. x  e.  a  Tr  x )  ->  a  e.  On )
109ralimi 2788 . . . . 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 3327 . . . 4  |-  ( N 
C_  On  <->  A. a  e.  N  a  e.  On )
1311, 12sylibr 205 . . 3  |-  ( ( Tr  N  /\  A. x  e.  N  Tr  x )  ->  N  C_  On )
14 ordon 4798 . . . 4  |-  Ord  On
1514a1i 11 . . 3  |-  ( ( Tr  N  /\  A. x  e.  N  Tr  x )  ->  Ord  On )
16 trssord 4633 . . 3  |-  ( ( Tr  N  /\  N  C_  On  /\  Ord  On )  ->  Ord  N )
177, 13, 15, 16syl3anc 1185 . 2  |-  ( ( Tr  N  /\  A. x  e.  N  Tr  x )  ->  Ord  N )
186, 17impbii 182 1  |-  ( Ord 
N  <->  ( Tr  N  /\  A. x  e.  N  Tr  x ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    /\ wa 360    e. wcel 1728   A.wral 2712    C_ wss 3309   Tr wtr 4333   Ord word 4615   Oncon0 4616
This theorem is referenced by:  dford4  27212
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1628  ax-9 1669  ax-8 1690  ax-13 1730  ax-14 1732  ax-6 1747  ax-7 1752  ax-11 1764  ax-12 1954  ax-ext 2424  ax-sep 4361  ax-nul 4369  ax-pr 4438  ax-un 4736  ax-reg 7596
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1661  df-eu 2292  df-mo 2293  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2717  df-rex 2718  df-rab 2721  df-v 2967  df-sbc 3171  df-dif 3312  df-un 3314  df-in 3316  df-ss 3323  df-pss 3325  df-nul 3617  df-if 3768  df-sn 3849  df-pr 3850  df-tp 3851  df-op 3852  df-uni 4045  df-br 4244  df-opab 4298  df-tr 4334  df-eprel 4529  df-po 4538  df-so 4539  df-fr 4576  df-we 4578  df-ord 4619  df-on 4620  df-suc 4622
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