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Theorem ordiso 7477
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 5180 . . . . 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 3029 . . . . 5  |-  ( (  _I  |`  A )  e.  _V  ->  ( (  _I  |`  A )  Isom  _E  ,  _E  ( A ,  A )  ->  E. f  f  Isom  _E  ,  _E  ( A ,  A ) ) )
51, 2, 4ee10 1385 . . . 4  |-  ( A  e.  On  ->  E. f 
f  Isom  _E  ,  _E  ( A ,  A ) )
65adantr 452 . . 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 1636 . . 3  |-  ( A  =  B  ->  ( E. f  f  Isom  _E  ,  _E  ( A ,  A )  <->  E. f 
f  Isom  _E  ,  _E  ( A ,  B ) ) )
96, 8syl5ibcom 212 . 2  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  =  B  ->  E. f  f  Isom  _E  ,  _E  ( A ,  B ) ) )
10 eloni 4583 . . . 4  |-  ( A  e.  On  ->  Ord  A )
11 eloni 4583 . . . 4  |-  ( B  e.  On  ->  Ord  B )
12 ordiso2 7476 . . . . . 6  |-  ( ( f  Isom  _E  ,  _E  ( A ,  B )  /\  Ord  A  /\  Ord  B )  ->  A  =  B )
13123coml 1160 . . . . 5  |-  ( ( Ord  A  /\  Ord  B  /\  f  Isom  _E  ,  _E  ( A ,  B
) )  ->  A  =  B )
14133expia 1155 . . . 4  |-  ( ( Ord  A  /\  Ord  B )  ->  ( f  Isom  _E  ,  _E  ( A ,  B )  ->  A  =  B ) )
1510, 11, 14syl2an 464 . . 3  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( f  Isom  _E  ,  _E  ( A ,  B
)  ->  A  =  B ) )
1615exlimdv 1646 . 2  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( E. f  f 
Isom  _E  ,  _E  ( A ,  B )  ->  A  =  B ) )
179, 16impbid 184 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    <-> wb 177    /\ wa 359   E.wex 1550    = wceq 1652    e. wcel 1725   _Vcvv 2948    _E cep 4484    _I cid 4485   Ord word 4572   Oncon0 4573    |` cres 4872    Isom wiso 5447
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-pow 4369  ax-pr 4395  ax-un 4693
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-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455
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