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Theorem finnisoeu 7994
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 2436 . . . . 5  |- OrdIso ( R ,  A )  = OrdIso
( R ,  A
)
21oiexg 7504 . . . 4  |-  ( A  e.  Fin  -> OrdIso ( R ,  A )  e. 
_V )
32adantl 453 . . 3  |-  ( ( R  Or  A  /\  A  e.  Fin )  -> OrdIso ( R ,  A
)  e.  _V )
4 simpr 448 . . . . 5  |-  ( ( R  Or  A  /\  A  e.  Fin )  ->  A  e.  Fin )
5 wofi 7356 . . . . 5  |-  ( ( R  Or  A  /\  A  e.  Fin )  ->  R  We  A )
61oiiso 7506 . . . . 5  |-  ( ( A  e.  Fin  /\  R  We  A )  -> OrdIso ( R ,  A
)  Isom  _E  ,  R  ( dom OrdIso ( R ,  A ) ,  A
) )
74, 5, 6syl2anc 643 . . . 4  |-  ( ( R  Or  A  /\  A  e.  Fin )  -> OrdIso ( R ,  A
)  Isom  _E  ,  R  ( dom OrdIso ( R ,  A ) ,  A
) )
81oien 7507 . . . . . . . 8  |-  ( ( A  e.  Fin  /\  R  We  A )  ->  dom OrdIso ( R ,  A )  ~~  A
)
94, 5, 8syl2anc 643 . . . . . . 7  |-  ( ( R  Or  A  /\  A  e.  Fin )  ->  dom OrdIso ( R ,  A )  ~~  A
)
10 ficardid 7849 . . . . . . . . 9  |-  ( A  e.  Fin  ->  ( card `  A )  ~~  A )
1110adantl 453 . . . . . . . 8  |-  ( ( R  Or  A  /\  A  e.  Fin )  ->  ( card `  A
)  ~~  A )
1211ensymd 7158 . . . . . . 7  |-  ( ( R  Or  A  /\  A  e.  Fin )  ->  A  ~~  ( card `  A ) )
13 entr 7159 . . . . . . 7  |-  ( ( dom OrdIso ( R ,  A )  ~~  A  /\  A  ~~  ( card `  A ) )  ->  dom OrdIso ( R ,  A
)  ~~  ( card `  A ) )
149, 12, 13syl2anc 643 . . . . . 6  |-  ( ( R  Or  A  /\  A  e.  Fin )  ->  dom OrdIso ( R ,  A )  ~~  ( card `  A ) )
151oion 7505 . . . . . . . 8  |-  ( A  e.  Fin  ->  dom OrdIso ( R ,  A )  e.  On )
1615adantl 453 . . . . . . 7  |-  ( ( R  Or  A  /\  A  e.  Fin )  ->  dom OrdIso ( R ,  A )  e.  On )
17 ficardom 7848 . . . . . . . 8  |-  ( A  e.  Fin  ->  ( card `  A )  e. 
om )
1817adantl 453 . . . . . . 7  |-  ( ( R  Or  A  /\  A  e.  Fin )  ->  ( card `  A
)  e.  om )
19 onomeneq 7296 . . . . . . 7  |-  ( ( dom OrdIso ( R ,  A )  e.  On  /\  ( card `  A
)  e.  om )  ->  ( dom OrdIso ( R ,  A )  ~~  ( card `  A )  <->  dom OrdIso ( R ,  A )  =  ( card `  A
) ) )
2016, 18, 19syl2anc 643 . . . . . 6  |-  ( ( R  Or  A  /\  A  e.  Fin )  ->  ( dom OrdIso ( R ,  A )  ~~  ( card `  A )  <->  dom OrdIso ( R ,  A )  =  ( card `  A
) ) )
2114, 20mpbid 202 . . . . 5  |-  ( ( R  Or  A  /\  A  e.  Fin )  ->  dom OrdIso ( R ,  A )  =  (
card `  A )
)
22 isoeq4 6042 . . . . 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 ) ) )
2321, 22syl 16 . . . 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
) ) )
247, 23mpbid 202 . . 3  |-  ( ( R  Or  A  /\  A  e.  Fin )  -> OrdIso ( R ,  A
)  Isom  _E  ,  R  ( ( card `  A
) ,  A ) )
25 isoeq1 6039 . . . 4  |-  ( f  = OrdIso ( R ,  A )  ->  (
f  Isom  _E  ,  R  ( ( card `  A
) ,  A )  <-> OrdIso ( R ,  A ) 
Isom  _E  ,  R  ( ( card `  A
) ,  A ) ) )
2625spcegv 3037 . . 3  |-  (OrdIso ( R ,  A )  e.  _V  ->  (OrdIso ( R ,  A )  Isom  _E  ,  R  ( ( card `  A
) ,  A )  ->  E. f  f  Isom  _E  ,  R  ( (
card `  A ) ,  A ) ) )
273, 24, 26sylc 58 . 2  |-  ( ( R  Or  A  /\  A  e.  Fin )  ->  E. f  f  Isom  _E  ,  R  ( (
card `  A ) ,  A ) )
28 wemoiso2 6079 . . 3  |-  ( R  We  A  ->  E* f  f  Isom  _E  ,  R  ( ( card `  A ) ,  A
) )
295, 28syl 16 . 2  |-  ( ( R  Or  A  /\  A  e.  Fin )  ->  E* f  f  Isom  _E  ,  R  ( (
card `  A ) ,  A ) )
30 eu5 2319 . 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 ) ) )
3127, 29, 30sylanbrc 646 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 177    /\ wa 359   E.wex 1550    = wceq 1652    e. wcel 1725   E!weu 2281   E*wmo 2282   _Vcvv 2956   class class class wbr 4212    _E cep 4492    Or wor 4502    We wwe 4540   Oncon0 4581   omcom 4845   dom cdm 4878   ` cfv 5454    Isom wiso 5455    ~~ cen 7106   Fincfn 7109  OrdIsocoi 7478   cardccrd 7822
This theorem is referenced by:  iunfictbso  7995
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 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-se 4542  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-isom 5463  df-riota 6549  df-recs 6633  df-1o 6724  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-oi 7479  df-card 7826
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