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Theorem ordiso 7066
Description: Order-isomorphic ordinal numbers are equal. (Contributed by Jeff Hankins, 16-Oct-2009.) (Proof shortened by Mario Carneiro, 24-Jun-2015.)
Assertion
Ref Expression
ordiso  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  =  B  <->  E. f  f  Isom  _E  ,  _E  ( A ,  B ) ) )
Distinct variable groups:    A, f    B, f

Proof of Theorem ordiso
StepHypRef Expression
1 resiexg 4970 . . . . 5  |-  ( A  e.  On  ->  (  _I  |`  A )  e. 
_V )
2 isoid 5832 . . . . 5  |-  (  _I  |`  A )  Isom  _E  ,  _E  ( A ,  A
)
3 isoeq1 5823 . . . . . 6  |-  ( f  =  (  _I  |`  A )  ->  ( f  Isom  _E  ,  _E  ( A ,  A )  <->  (  _I  |`  A )  Isom  _E  ,  _E  ( A ,  A
) ) )
43spcegv 2840 . . . . 5  |-  ( (  _I  |`  A )  e.  _V  ->  ( (  _I  |`  A )  Isom  _E  ,  _E  ( A ,  A )  ->  E. f  f  Isom  _E  ,  _E  ( A ,  A ) ) )
51, 2, 4mpisyl 1457 . . . 4  |-  ( A  e.  On  ->  E. f 
f  Isom  _E  ,  _E  ( A ,  A ) )
65adantr 276 . . 3  |-  ( ( A  e.  On  /\  B  e.  On )  ->  E. f  f  Isom  _E  ,  _E  ( A ,  A ) )
7 isoeq5 5827 . . . 4  |-  ( A  =  B  ->  (
f  Isom  _E  ,  _E  ( A ,  A )  <-> 
f  Isom  _E  ,  _E  ( A ,  B ) ) )
87exbidv 1836 . . 3  |-  ( A  =  B  ->  ( E. f  f  Isom  _E  ,  _E  ( A ,  A )  <->  E. f 
f  Isom  _E  ,  _E  ( A ,  B ) ) )
96, 8syl5ibcom 155 . 2  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  =  B  ->  E. f  f  Isom  _E  ,  _E  ( A ,  B ) ) )
10 eloni 4393 . . . 4  |-  ( A  e.  On  ->  Ord  A )
11 eloni 4393 . . . 4  |-  ( B  e.  On  ->  Ord  B )
12 ordiso2 7065 . . . . . 6  |-  ( ( f  Isom  _E  ,  _E  ( A ,  B )  /\  Ord  A  /\  Ord  B )  ->  A  =  B )
13123coml 1212 . . . . 5  |-  ( ( Ord  A  /\  Ord  B  /\  f  Isom  _E  ,  _E  ( A ,  B
) )  ->  A  =  B )
14133expia 1207 . . . 4  |-  ( ( Ord  A  /\  Ord  B )  ->  ( f  Isom  _E  ,  _E  ( A ,  B )  ->  A  =  B ) )
1510, 11, 14syl2an 289 . . 3  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( f  Isom  _E  ,  _E  ( A ,  B
)  ->  A  =  B ) )
1615exlimdv 1830 . 2  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( E. f  f 
Isom  _E  ,  _E  ( A ,  B )  ->  A  =  B ) )
179, 16impbid 129 1  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  =  B  <->  E. f  f  Isom  _E  ,  _E  ( A ,  B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    = wceq 1364   E.wex 1503    e. wcel 2160   _Vcvv 2752    _E cep 4305    _I cid 4306   Ord word 4380   Oncon0 4381    |` cres 4646    Isom wiso 5236
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2754  df-sbc 2978  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-eprel 4307  df-id 4311  df-iord 4384  df-on 4386  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-isom 5244
This theorem is referenced by: (None)
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