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Theorem dford3 27224
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 4422 . . 3  |-  ( Ord 
N  ->  Tr  N
)
2 ordelord 4430 . . . . 5  |-  ( ( Ord  N  /\  x  e.  N )  ->  Ord  x )
3 ordtr 4422 . . . . 5  |-  ( Ord  x  ->  Tr  x
)
42, 3syl 15 . . . 4  |-  ( ( Ord  N  /\  x  e.  N )  ->  Tr  x )
54ralrimiva 2639 . . 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 27222 . . . . 5  |-  ( ( Tr  N  /\  A. x  e.  N  Tr  x )  ->  A. a  e.  N  ( Tr  a  /\  A. x  e.  a  Tr  x ) )
9 dford3lem2 27223 . . . . . 6  |-  ( ( Tr  a  /\  A. x  e.  a  Tr  x )  ->  a  e.  On )
109ralimi 2631 . . . . 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 3183 . . . 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 4590 . . . 4  |-  Ord  On
1514a1i 10 . . 3  |-  ( ( Tr  N  /\  A. x  e.  N  Tr  x )  ->  Ord  On )
16 trssord 4425 . . 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 1696   A.wral 2556    C_ wss 3165   Tr wtr 4129   Ord word 4407   Oncon0 4408
This theorem is referenced by:  dford4  27225
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230  ax-un 4528  ax-reg 7322
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-tr 4130  df-eprel 4321  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-suc 4414
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