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Theorem finnisoeu 7740
Description: A finite totally ordered set has a unique order isomorphism to a finite ordinal. (Contributed by Stefan O'Rear, 16-Nov-2014.) (Proof shortened by Mario Carneiro, 26-Jun-2015.)
Assertion
Ref Expression
finnisoeu  |-  ( ( R  Or  A  /\  A  e.  Fin )  ->  E! f  f  Isom  _E  ,  R  ( (
card `  A ) ,  A ) )
Distinct variable groups:    R, f    A, f

Proof of Theorem finnisoeu
StepHypRef Expression
1 eqid 2283 . . . . 5  |- OrdIso ( R ,  A )  = OrdIso
( R ,  A
)
21oiexg 7250 . . . 4  |-  ( A  e.  Fin  -> OrdIso ( R ,  A )  e. 
_V )
32adantl 452 . . 3  |-  ( ( R  Or  A  /\  A  e.  Fin )  -> OrdIso ( R ,  A
)  e.  _V )
4 simpr 447 . . . . 5  |-  ( ( R  Or  A  /\  A  e.  Fin )  ->  A  e.  Fin )
5 wofi 7106 . . . . 5  |-  ( ( R  Or  A  /\  A  e.  Fin )  ->  R  We  A )
61oiiso 7252 . . . . 5  |-  ( ( A  e.  Fin  /\  R  We  A )  -> OrdIso ( R ,  A
)  Isom  _E  ,  R  ( dom OrdIso ( R ,  A ) ,  A
) )
74, 5, 6syl2anc 642 . . . 4  |-  ( ( R  Or  A  /\  A  e.  Fin )  -> OrdIso ( R ,  A
)  Isom  _E  ,  R  ( dom OrdIso ( R ,  A ) ,  A
) )
81oien 7253 . . . . . . . 8  |-  ( ( A  e.  Fin  /\  R  We  A )  ->  dom OrdIso ( R ,  A )  ~~  A
)
94, 5, 8syl2anc 642 . . . . . . 7  |-  ( ( R  Or  A  /\  A  e.  Fin )  ->  dom OrdIso ( R ,  A )  ~~  A
)
10 ficardid 7595 . . . . . . . . 9  |-  ( A  e.  Fin  ->  ( card `  A )  ~~  A )
1110adantl 452 . . . . . . . 8  |-  ( ( R  Or  A  /\  A  e.  Fin )  ->  ( card `  A
)  ~~  A )
12 ensym 6910 . . . . . . . 8  |-  ( (
card `  A )  ~~  A  ->  A  ~~  ( card `  A )
)
1311, 12syl 15 . . . . . . 7  |-  ( ( R  Or  A  /\  A  e.  Fin )  ->  A  ~~  ( card `  A ) )
14 entr 6913 . . . . . . 7  |-  ( ( dom OrdIso ( R ,  A )  ~~  A  /\  A  ~~  ( card `  A ) )  ->  dom OrdIso ( R ,  A
)  ~~  ( card `  A ) )
159, 13, 14syl2anc 642 . . . . . 6  |-  ( ( R  Or  A  /\  A  e.  Fin )  ->  dom OrdIso ( R ,  A )  ~~  ( card `  A ) )
161oion 7251 . . . . . . . 8  |-  ( A  e.  Fin  ->  dom OrdIso ( R ,  A )  e.  On )
1716adantl 452 . . . . . . 7  |-  ( ( R  Or  A  /\  A  e.  Fin )  ->  dom OrdIso ( R ,  A )  e.  On )
18 ficardom 7594 . . . . . . . 8  |-  ( A  e.  Fin  ->  ( card `  A )  e. 
om )
1918adantl 452 . . . . . . 7  |-  ( ( R  Or  A  /\  A  e.  Fin )  ->  ( card `  A
)  e.  om )
20 onomeneq 7050 . . . . . . 7  |-  ( ( dom OrdIso ( R ,  A )  e.  On  /\  ( card `  A
)  e.  om )  ->  ( dom OrdIso ( R ,  A )  ~~  ( card `  A )  <->  dom OrdIso ( R ,  A )  =  ( card `  A
) ) )
2117, 19, 20syl2anc 642 . . . . . 6  |-  ( ( R  Or  A  /\  A  e.  Fin )  ->  ( dom OrdIso ( R ,  A )  ~~  ( card `  A )  <->  dom OrdIso ( R ,  A )  =  ( card `  A
) ) )
2215, 21mpbid 201 . . . . 5  |-  ( ( R  Or  A  /\  A  e.  Fin )  ->  dom OrdIso ( R ,  A )  =  (
card `  A )
)
23 isoeq4 5819 . . . . 5  |-  ( dom OrdIso ( R ,  A )  =  ( card `  A
)  ->  (OrdIso ( R ,  A )  Isom  _E  ,  R  ( dom OrdIso ( R ,  A ) ,  A
)  <-> OrdIso ( R ,  A
)  Isom  _E  ,  R  ( ( card `  A
) ,  A ) ) )
2422, 23syl 15 . . . 4  |-  ( ( R  Or  A  /\  A  e.  Fin )  ->  (OrdIso ( R ,  A )  Isom  _E  ,  R  ( dom OrdIso ( R ,  A ) ,  A )  <-> OrdIso ( R ,  A )  Isom  _E  ,  R  ( ( card `  A ) ,  A
) ) )
257, 24mpbid 201 . . 3  |-  ( ( R  Or  A  /\  A  e.  Fin )  -> OrdIso ( R ,  A
)  Isom  _E  ,  R  ( ( card `  A
) ,  A ) )
26 isoeq1 5816 . . . 4  |-  ( f  = OrdIso ( R ,  A )  ->  (
f  Isom  _E  ,  R  ( ( card `  A
) ,  A )  <-> OrdIso ( R ,  A ) 
Isom  _E  ,  R  ( ( card `  A
) ,  A ) ) )
2726spcegv 2869 . . 3  |-  (OrdIso ( R ,  A )  e.  _V  ->  (OrdIso ( R ,  A )  Isom  _E  ,  R  ( ( card `  A
) ,  A )  ->  E. f  f  Isom  _E  ,  R  ( (
card `  A ) ,  A ) ) )
283, 25, 27sylc 56 . 2  |-  ( ( R  Or  A  /\  A  e.  Fin )  ->  E. f  f  Isom  _E  ,  R  ( (
card `  A ) ,  A ) )
29 wemoiso2 5856 . . 3  |-  ( R  We  A  ->  E* f  f  Isom  _E  ,  R  ( ( card `  A ) ,  A
) )
305, 29syl 15 . 2  |-  ( ( R  Or  A  /\  A  e.  Fin )  ->  E* f  f  Isom  _E  ,  R  ( (
card `  A ) ,  A ) )
31 eu5 2181 . 2  |-  ( E! f  f  Isom  _E  ,  R  ( ( card `  A ) ,  A
)  <->  ( E. f 
f  Isom  _E  ,  R  ( ( card `  A
) ,  A )  /\  E* f  f 
Isom  _E  ,  R  ( ( card `  A
) ,  A ) ) )
3228, 30, 31sylanbrc 645 1  |-  ( ( R  Or  A  /\  A  e.  Fin )  ->  E! f  f  Isom  _E  ,  R  ( (
card `  A ) ,  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   E.wex 1528    = wceq 1623    e. wcel 1684   E!weu 2143   E*wmo 2144   _Vcvv 2788   class class class wbr 4023    _E cep 4303    Or wor 4313    We wwe 4351   Oncon0 4392   omcom 4656   dom cdm 4689   ` cfv 5255    Isom wiso 5256    ~~ cen 6860   Fincfn 6863  OrdIsocoi 7224   cardccrd 7568
This theorem is referenced by:  iunfictbso  7741
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-riota 6304  df-recs 6388  df-1o 6479  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-oi 7225  df-card 7572
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