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Theorem ordiso 7751
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 6535 . . . . 5  |-  ( A  e.  On  ->  (  _I  |`  A )  e. 
_V )
2 isoid 6041 . . . . 5  |-  (  _I  |`  A )  Isom  _E  ,  _E  ( A ,  A
)
3 isoeq1 6031 . . . . . 6  |-  ( f  =  (  _I  |`  A )  ->  ( f  Isom  _E  ,  _E  ( A ,  A )  <->  (  _I  |`  A )  Isom  _E  ,  _E  ( A ,  A
) ) )
43spcegv 3079 . . . . 5  |-  ( (  _I  |`  A )  e.  _V  ->  ( (  _I  |`  A )  Isom  _E  ,  _E  ( A ,  A )  ->  E. f  f  Isom  _E  ,  _E  ( A ,  A ) ) )
51, 2, 4mpisyl 18 . . . 4  |-  ( A  e.  On  ->  E. f 
f  Isom  _E  ,  _E  ( A ,  A ) )
65adantr 465 . . 3  |-  ( ( A  e.  On  /\  B  e.  On )  ->  E. f  f  Isom  _E  ,  _E  ( A ,  A ) )
7 isoeq5 6035 . . . 4  |-  ( A  =  B  ->  (
f  Isom  _E  ,  _E  ( A ,  A )  <-> 
f  Isom  _E  ,  _E  ( A ,  B ) ) )
87exbidv 1680 . . 3  |-  ( A  =  B  ->  ( E. f  f  Isom  _E  ,  _E  ( A ,  A )  <->  E. f 
f  Isom  _E  ,  _E  ( A ,  B ) ) )
96, 8syl5ibcom 220 . 2  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  =  B  ->  E. f  f  Isom  _E  ,  _E  ( A ,  B ) ) )
10 eloni 4750 . . . 4  |-  ( A  e.  On  ->  Ord  A )
11 eloni 4750 . . . 4  |-  ( B  e.  On  ->  Ord  B )
12 ordiso2 7750 . . . . . 6  |-  ( ( f  Isom  _E  ,  _E  ( A ,  B )  /\  Ord  A  /\  Ord  B )  ->  A  =  B )
13123coml 1194 . . . . 5  |-  ( ( Ord  A  /\  Ord  B  /\  f  Isom  _E  ,  _E  ( A ,  B
) )  ->  A  =  B )
14133expia 1189 . . . 4  |-  ( ( Ord  A  /\  Ord  B )  ->  ( f  Isom  _E  ,  _E  ( A ,  B )  ->  A  =  B ) )
1510, 11, 14syl2an 477 . . 3  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( f  Isom  _E  ,  _E  ( A ,  B
)  ->  A  =  B ) )
1615exlimdv 1690 . 2  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( E. f  f 
Isom  _E  ,  _E  ( A ,  B )  ->  A  =  B ) )
179, 16impbid 191 1  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  =  B  <->  E. f  f  Isom  _E  ,  _E  ( A ,  B ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369   E.wex 1586    e. wcel 1756   _Vcvv 2993    _E cep 4651    _I cid 4652   Ord word 4739   Oncon0 4740    |` cres 4863    Isom wiso 5440
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2741  df-rex 2742  df-rab 2745  df-v 2995  df-sbc 3208  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-pss 3365  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-tp 3903  df-op 3905  df-uni 4113  df-br 4314  df-opab 4372  df-mpt 4373  df-tr 4407  df-eprel 4653  df-id 4657  df-po 4662  df-so 4663  df-fr 4700  df-we 4702  df-ord 4743  df-on 4744  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-isom 5448
This theorem is referenced by: (None)
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