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Theorem ordiso 7975
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 6720 . . . . 5  |-  ( A  e.  On  ->  (  _I  |`  A )  e. 
_V )
2 isoid 6208 . . . . 5  |-  (  _I  |`  A )  Isom  _E  ,  _E  ( A ,  A
)
3 isoeq1 6198 . . . . . 6  |-  ( f  =  (  _I  |`  A )  ->  ( f  Isom  _E  ,  _E  ( A ,  A )  <->  (  _I  |`  A )  Isom  _E  ,  _E  ( A ,  A
) ) )
43spcegv 3145 . . . . 5  |-  ( (  _I  |`  A )  e.  _V  ->  ( (  _I  |`  A )  Isom  _E  ,  _E  ( A ,  A )  ->  E. f  f  Isom  _E  ,  _E  ( A ,  A ) ) )
51, 2, 4mpisyl 21 . . . 4  |-  ( A  e.  On  ->  E. f 
f  Isom  _E  ,  _E  ( A ,  A ) )
65adantr 463 . . 3  |-  ( ( A  e.  On  /\  B  e.  On )  ->  E. f  f  Isom  _E  ,  _E  ( A ,  A ) )
7 isoeq5 6202 . . . 4  |-  ( A  =  B  ->  (
f  Isom  _E  ,  _E  ( A ,  A )  <-> 
f  Isom  _E  ,  _E  ( A ,  B ) ) )
87exbidv 1735 . . 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 5420 . . . 4  |-  ( A  e.  On  ->  Ord  A )
11 eloni 5420 . . . 4  |-  ( B  e.  On  ->  Ord  B )
12 ordiso2 7974 . . . . . 6  |-  ( ( f  Isom  _E  ,  _E  ( A ,  B )  /\  Ord  A  /\  Ord  B )  ->  A  =  B )
13123coml 1204 . . . . 5  |-  ( ( Ord  A  /\  Ord  B  /\  f  Isom  _E  ,  _E  ( A ,  B
) )  ->  A  =  B )
14133expia 1199 . . . 4  |-  ( ( Ord  A  /\  Ord  B )  ->  ( f  Isom  _E  ,  _E  ( A ,  B )  ->  A  =  B ) )
1510, 11, 14syl2an 475 . . 3  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( f  Isom  _E  ,  _E  ( A ,  B
)  ->  A  =  B ) )
1615exlimdv 1745 . 2  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( E. f  f 
Isom  _E  ,  _E  ( A ,  B )  ->  A  =  B ) )
179, 16impbid 190 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 367    = wceq 1405   E.wex 1633    e. wcel 1842   _Vcvv 3059    _E cep 4732    _I cid 4733    |` cres 4825   Ord word 5409   Oncon0 5410    Isom wiso 5570
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-rab 2763  df-v 3061  df-sbc 3278  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4192  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-ord 5413  df-on 5414  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-isom 5578
This theorem is referenced by: (None)
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