MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  finnisoeu Structured version   Unicode version

Theorem finnisoeu 8533
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 2420 . . . . 5  |- OrdIso ( R ,  A )  = OrdIso
( R ,  A
)
21oiexg 8041 . . . 4  |-  ( A  e.  Fin  -> OrdIso ( R ,  A )  e. 
_V )
32adantl 467 . . 3  |-  ( ( R  Or  A  /\  A  e.  Fin )  -> OrdIso ( R ,  A
)  e.  _V )
4 simpr 462 . . . . 5  |-  ( ( R  Or  A  /\  A  e.  Fin )  ->  A  e.  Fin )
5 wofi 7817 . . . . 5  |-  ( ( R  Or  A  /\  A  e.  Fin )  ->  R  We  A )
61oiiso 8043 . . . . 5  |-  ( ( A  e.  Fin  /\  R  We  A )  -> OrdIso ( R ,  A
)  Isom  _E  ,  R  ( dom OrdIso ( R ,  A ) ,  A
) )
74, 5, 6syl2anc 665 . . . 4  |-  ( ( R  Or  A  /\  A  e.  Fin )  -> OrdIso ( R ,  A
)  Isom  _E  ,  R  ( dom OrdIso ( R ,  A ) ,  A
) )
81oien 8044 . . . . . . . 8  |-  ( ( A  e.  Fin  /\  R  We  A )  ->  dom OrdIso ( R ,  A )  ~~  A
)
94, 5, 8syl2anc 665 . . . . . . 7  |-  ( ( R  Or  A  /\  A  e.  Fin )  ->  dom OrdIso ( R ,  A )  ~~  A
)
10 ficardid 8386 . . . . . . . . 9  |-  ( A  e.  Fin  ->  ( card `  A )  ~~  A )
1110adantl 467 . . . . . . . 8  |-  ( ( R  Or  A  /\  A  e.  Fin )  ->  ( card `  A
)  ~~  A )
1211ensymd 7618 . . . . . . 7  |-  ( ( R  Or  A  /\  A  e.  Fin )  ->  A  ~~  ( card `  A ) )
13 entr 7619 . . . . . . 7  |-  ( ( dom OrdIso ( R ,  A )  ~~  A  /\  A  ~~  ( card `  A ) )  ->  dom OrdIso ( R ,  A
)  ~~  ( card `  A ) )
149, 12, 13syl2anc 665 . . . . . 6  |-  ( ( R  Or  A  /\  A  e.  Fin )  ->  dom OrdIso ( R ,  A )  ~~  ( card `  A ) )
151oion 8042 . . . . . . . 8  |-  ( A  e.  Fin  ->  dom OrdIso ( R ,  A )  e.  On )
1615adantl 467 . . . . . . 7  |-  ( ( R  Or  A  /\  A  e.  Fin )  ->  dom OrdIso ( R ,  A )  e.  On )
17 ficardom 8385 . . . . . . . 8  |-  ( A  e.  Fin  ->  ( card `  A )  e. 
om )
1817adantl 467 . . . . . . 7  |-  ( ( R  Or  A  /\  A  e.  Fin )  ->  ( card `  A
)  e.  om )
19 onomeneq 7759 . . . . . . 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 665 . . . . . 6  |-  ( ( R  Or  A  /\  A  e.  Fin )  ->  ( dom OrdIso ( R ,  A )  ~~  ( card `  A )  <->  dom OrdIso ( R ,  A )  =  ( card `  A
) ) )
2114, 20mpbid 213 . . . . 5  |-  ( ( R  Or  A  /\  A  e.  Fin )  ->  dom OrdIso ( R ,  A )  =  (
card `  A )
)
22 isoeq4 6219 . . . . 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 17 . . . 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 213 . . 3  |-  ( ( R  Or  A  /\  A  e.  Fin )  -> OrdIso ( R ,  A
)  Isom  _E  ,  R  ( ( card `  A
) ,  A ) )
25 isoeq1 6216 . . . 4  |-  ( f  = OrdIso ( R ,  A )  ->  (
f  Isom  _E  ,  R  ( ( card `  A
) ,  A )  <-> OrdIso ( R ,  A ) 
Isom  _E  ,  R  ( ( card `  A
) ,  A ) ) )
2625spcegv 3164 . . 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 62 . 2  |-  ( ( R  Or  A  /\  A  e.  Fin )  ->  E. f  f  Isom  _E  ,  R  ( (
card `  A ) ,  A ) )
28 wemoiso2 6784 . . 3  |-  ( R  We  A  ->  E* f  f  Isom  _E  ,  R  ( ( card `  A ) ,  A
) )
295, 28syl 17 . 2  |-  ( ( R  Or  A  /\  A  e.  Fin )  ->  E* f  f  Isom  _E  ,  R  ( (
card `  A ) ,  A ) )
30 eu5 2290 . 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 668 1  |-  ( ( R  Or  A  /\  A  e.  Fin )  ->  E! f  f  Isom  _E  ,  R  ( (
card `  A ) ,  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    = wceq 1437   E.wex 1659    e. wcel 1867   E!weu 2263   E*wmo 2264   _Vcvv 3078   class class class wbr 4417    _E cep 4754    Or wor 4765    We wwe 4803   dom cdm 4845   Oncon0 5433   ` cfv 5592    Isom wiso 5593   omcom 6697    ~~ cen 7565   Fincfn 7568  OrdIsocoi 8015   cardccrd 8359
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-rep 4529  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-reu 2780  df-rmo 2781  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-pss 3449  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-tp 3998  df-op 4000  df-uni 4214  df-int 4250  df-iun 4295  df-br 4418  df-opab 4476  df-mpt 4477  df-tr 4512  df-eprel 4756  df-id 4760  df-po 4766  df-so 4767  df-fr 4804  df-se 4805  df-we 4806  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-pred 5390  df-ord 5436  df-on 5437  df-lim 5438  df-suc 5439  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-isom 5601  df-riota 6258  df-om 6698  df-wrecs 7027  df-recs 7089  df-1o 7181  df-er 7362  df-en 7569  df-dom 7570  df-sdom 7571  df-fin 7572  df-oi 8016  df-card 8363
This theorem is referenced by:  iunfictbso  8534
  Copyright terms: Public domain W3C validator