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Theorem ordiso 6729
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 4757 . . . . 5  |-  ( A  e.  On  ->  (  _I  |`  A )  e. 
_V )
2 isoid 5589 . . . . 5  |-  (  _I  |`  A )  Isom  _E  ,  _E  ( A ,  A
)
3 isoeq1 5580 . . . . . 6  |-  ( f  =  (  _I  |`  A )  ->  ( f  Isom  _E  ,  _E  ( A ,  A )  <->  (  _I  |`  A )  Isom  _E  ,  _E  ( A ,  A
) ) )
43spcegv 2707 . . . . 5  |-  ( (  _I  |`  A )  e.  _V  ->  ( (  _I  |`  A )  Isom  _E  ,  _E  ( A ,  A )  ->  E. f  f  Isom  _E  ,  _E  ( A ,  A ) ) )
51, 2, 4mpisyl 1380 . . . 4  |-  ( A  e.  On  ->  E. f 
f  Isom  _E  ,  _E  ( A ,  A ) )
65adantr 270 . . 3  |-  ( ( A  e.  On  /\  B  e.  On )  ->  E. f  f  Isom  _E  ,  _E  ( A ,  A ) )
7 isoeq5 5584 . . . 4  |-  ( A  =  B  ->  (
f  Isom  _E  ,  _E  ( A ,  A )  <-> 
f  Isom  _E  ,  _E  ( A ,  B ) ) )
87exbidv 1753 . . 3  |-  ( A  =  B  ->  ( E. f  f  Isom  _E  ,  _E  ( A ,  A )  <->  E. f 
f  Isom  _E  ,  _E  ( A ,  B ) ) )
96, 8syl5ibcom 153 . 2  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  =  B  ->  E. f  f  Isom  _E  ,  _E  ( A ,  B ) ) )
10 eloni 4202 . . . 4  |-  ( A  e.  On  ->  Ord  A )
11 eloni 4202 . . . 4  |-  ( B  e.  On  ->  Ord  B )
12 ordiso2 6728 . . . . . 6  |-  ( ( f  Isom  _E  ,  _E  ( A ,  B )  /\  Ord  A  /\  Ord  B )  ->  A  =  B )
13123coml 1150 . . . . 5  |-  ( ( Ord  A  /\  Ord  B  /\  f  Isom  _E  ,  _E  ( A ,  B
) )  ->  A  =  B )
14133expia 1145 . . . 4  |-  ( ( Ord  A  /\  Ord  B )  ->  ( f  Isom  _E  ,  _E  ( A ,  B )  ->  A  =  B ) )
1510, 11, 14syl2an 283 . . 3  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( f  Isom  _E  ,  _E  ( A ,  B
)  ->  A  =  B ) )
1615exlimdv 1747 . 2  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( E. f  f 
Isom  _E  ,  _E  ( A ,  B )  ->  A  =  B ) )
179, 16impbid 127 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 102    <-> wb 103    = wceq 1289   E.wex 1426    e. wcel 1438   _Vcvv 2619    _E cep 4114    _I cid 4115   Ord word 4189   Oncon0 4190    |` cres 4440    Isom wiso 5016
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036  ax-un 4260  ax-setind 4353
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-rab 2368  df-v 2621  df-sbc 2841  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-opab 3900  df-mpt 3901  df-tr 3937  df-eprel 4116  df-id 4120  df-iord 4193  df-on 4195  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-f1 5020  df-fo 5021  df-f1o 5022  df-fv 5023  df-isom 5024
This theorem is referenced by: (None)
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