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Theorem dford3 27036
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 4587 . . 3  |-  ( Ord 
N  ->  Tr  N
)
2 ordelord 4595 . . . . 5  |-  ( ( Ord  N  /\  x  e.  N )  ->  Ord  x )
3 ordtr 4587 . . . . 5  |-  ( Ord  x  ->  Tr  x
)
42, 3syl 16 . . . 4  |-  ( ( Ord  N  /\  x  e.  N )  ->  Tr  x )
54ralrimiva 2781 . . 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 27034 . . . . 5  |-  ( ( Tr  N  /\  A. x  e.  N  Tr  x )  ->  A. a  e.  N  ( Tr  a  /\  A. x  e.  a  Tr  x ) )
9 dford3lem2 27035 . . . . . 6  |-  ( ( Tr  a  /\  A. x  e.  a  Tr  x )  ->  a  e.  On )
109ralimi 2773 . . . . 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 3330 . . . 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 4754 . . . 4  |-  Ord  On
1514a1i 11 . . 3  |-  ( ( Tr  N  /\  A. x  e.  N  Tr  x )  ->  Ord  On )
16 trssord 4590 . . 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 1725   A.wral 2697    C_ wss 3312   Tr wtr 4294   Ord word 4572   Oncon0 4573
This theorem is referenced by:  dford4  27037
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395  ax-un 4692  ax-reg 7549
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-tr 4295  df-eprel 4486  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-suc 4579
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