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Theorem finnisoeu 8495
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 2467 . . . . 5  |- OrdIso ( R ,  A )  = OrdIso
( R ,  A
)
21oiexg 7961 . . . 4  |-  ( A  e.  Fin  -> OrdIso ( R ,  A )  e. 
_V )
32adantl 466 . . 3  |-  ( ( R  Or  A  /\  A  e.  Fin )  -> OrdIso ( R ,  A
)  e.  _V )
4 simpr 461 . . . . 5  |-  ( ( R  Or  A  /\  A  e.  Fin )  ->  A  e.  Fin )
5 wofi 7770 . . . . 5  |-  ( ( R  Or  A  /\  A  e.  Fin )  ->  R  We  A )
61oiiso 7963 . . . . 5  |-  ( ( A  e.  Fin  /\  R  We  A )  -> OrdIso ( R ,  A
)  Isom  _E  ,  R  ( dom OrdIso ( R ,  A ) ,  A
) )
74, 5, 6syl2anc 661 . . . 4  |-  ( ( R  Or  A  /\  A  e.  Fin )  -> OrdIso ( R ,  A
)  Isom  _E  ,  R  ( dom OrdIso ( R ,  A ) ,  A
) )
81oien 7964 . . . . . . . 8  |-  ( ( A  e.  Fin  /\  R  We  A )  ->  dom OrdIso ( R ,  A )  ~~  A
)
94, 5, 8syl2anc 661 . . . . . . 7  |-  ( ( R  Or  A  /\  A  e.  Fin )  ->  dom OrdIso ( R ,  A )  ~~  A
)
10 ficardid 8344 . . . . . . . . 9  |-  ( A  e.  Fin  ->  ( card `  A )  ~~  A )
1110adantl 466 . . . . . . . 8  |-  ( ( R  Or  A  /\  A  e.  Fin )  ->  ( card `  A
)  ~~  A )
1211ensymd 7567 . . . . . . 7  |-  ( ( R  Or  A  /\  A  e.  Fin )  ->  A  ~~  ( card `  A ) )
13 entr 7568 . . . . . . 7  |-  ( ( dom OrdIso ( R ,  A )  ~~  A  /\  A  ~~  ( card `  A ) )  ->  dom OrdIso ( R ,  A
)  ~~  ( card `  A ) )
149, 12, 13syl2anc 661 . . . . . 6  |-  ( ( R  Or  A  /\  A  e.  Fin )  ->  dom OrdIso ( R ,  A )  ~~  ( card `  A ) )
151oion 7962 . . . . . . . 8  |-  ( A  e.  Fin  ->  dom OrdIso ( R ,  A )  e.  On )
1615adantl 466 . . . . . . 7  |-  ( ( R  Or  A  /\  A  e.  Fin )  ->  dom OrdIso ( R ,  A )  e.  On )
17 ficardom 8343 . . . . . . . 8  |-  ( A  e.  Fin  ->  ( card `  A )  e. 
om )
1817adantl 466 . . . . . . 7  |-  ( ( R  Or  A  /\  A  e.  Fin )  ->  ( card `  A
)  e.  om )
19 onomeneq 7708 . . . . . . 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 661 . . . . . 6  |-  ( ( R  Or  A  /\  A  e.  Fin )  ->  ( dom OrdIso ( R ,  A )  ~~  ( card `  A )  <->  dom OrdIso ( R ,  A )  =  ( card `  A
) ) )
2114, 20mpbid 210 . . . . 5  |-  ( ( R  Or  A  /\  A  e.  Fin )  ->  dom OrdIso ( R ,  A )  =  (
card `  A )
)
22 isoeq4 6207 . . . . 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 210 . . 3  |-  ( ( R  Or  A  /\  A  e.  Fin )  -> OrdIso ( R ,  A
)  Isom  _E  ,  R  ( ( card `  A
) ,  A ) )
25 isoeq1 6204 . . . 4  |-  ( f  = OrdIso ( R ,  A )  ->  (
f  Isom  _E  ,  R  ( ( card `  A
) ,  A )  <-> OrdIso ( R ,  A ) 
Isom  _E  ,  R  ( ( card `  A
) ,  A ) ) )
2625spcegv 3199 . . 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 60 . 2  |-  ( ( R  Or  A  /\  A  e.  Fin )  ->  E. f  f  Isom  _E  ,  R  ( (
card `  A ) ,  A ) )
28 wemoiso2 6771 . . 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 2305 . 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 664 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 184    /\ wa 369    = wceq 1379   E.wex 1596    e. wcel 1767   E!weu 2275   E*wmo 2276   _Vcvv 3113   class class class wbr 4447    _E cep 4789    Or wor 4799    We wwe 4837   Oncon0 4878   dom cdm 4999   ` cfv 5588    Isom wiso 5589   omcom 6685    ~~ cen 7514   Fincfn 7517  OrdIsocoi 7935   cardccrd 8317
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-isom 5597  df-riota 6246  df-om 6686  df-recs 7043  df-1o 7131  df-er 7312  df-en 7518  df-dom 7519  df-sdom 7520  df-fin 7521  df-oi 7936  df-card 8321
This theorem is referenced by:  iunfictbso  8496
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