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Theorem wofib 8085
Description: The only sets which are well-ordered forwards and backwards are finite sets. (Contributed by Mario Carneiro, 30-Jan-2014.) (Revised by Mario Carneiro, 23-May-2015.)
Hypothesis
Ref Expression
wofib.1  |-  A  e. 
_V
Assertion
Ref Expression
wofib  |-  ( ( R  Or  A  /\  A  e.  Fin )  <->  ( R  We  A  /\  `' R  We  A
) )

Proof of Theorem wofib
Dummy variables  y 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 wofi 7845 . . 3  |-  ( ( R  Or  A  /\  A  e.  Fin )  ->  R  We  A )
2 cnvso 5393 . . . 4  |-  ( R  Or  A  <->  `' R  Or  A )
3 wofi 7845 . . . 4  |-  ( ( `' R  Or  A  /\  A  e.  Fin )  ->  `' R  We  A )
42, 3sylanb 479 . . 3  |-  ( ( R  Or  A  /\  A  e.  Fin )  ->  `' R  We  A
)
51, 4jca 539 . 2  |-  ( ( R  Or  A  /\  A  e.  Fin )  ->  ( R  We  A  /\  `' R  We  A
) )
6 weso 4843 . . . 4  |-  ( R  We  A  ->  R  Or  A )
76adantr 471 . . 3  |-  ( ( R  We  A  /\  `' R  We  A
)  ->  R  Or  A )
8 peano2 6739 . . . . . . . . 9  |-  ( y  e.  om  ->  suc  y  e.  om )
9 sucidg 5519 . . . . . . . . 9  |-  ( y  e.  om  ->  y  e.  suc  y )
10 vex 3059 . . . . . . . . . . . . 13  |-  z  e. 
_V
11 vex 3059 . . . . . . . . . . . . 13  |-  y  e. 
_V
1210, 11brcnv 5035 . . . . . . . . . . . 12  |-  ( z `'  _E  y  <->  y  _E  z )
13 epel 4766 . . . . . . . . . . . 12  |-  ( y  _E  z  <->  y  e.  z )
1412, 13bitri 257 . . . . . . . . . . 11  |-  ( z `'  _E  y  <->  y  e.  z )
15 eleq2 2528 . . . . . . . . . . 11  |-  ( z  =  suc  y  -> 
( y  e.  z  <-> 
y  e.  suc  y
) )
1614, 15syl5bb 265 . . . . . . . . . 10  |-  ( z  =  suc  y  -> 
( z `'  _E  y 
<->  y  e.  suc  y
) )
1716rspcev 3161 . . . . . . . . 9  |-  ( ( suc  y  e.  om  /\  y  e.  suc  y
)  ->  E. z  e.  om  z `'  _E  y )
188, 9, 17syl2anc 671 . . . . . . . 8  |-  ( y  e.  om  ->  E. z  e.  om  z `'  _E  y )
19 dfrex2 2849 . . . . . . . 8  |-  ( E. z  e.  om  z `'  _E  y  <->  -.  A. z  e.  om  -.  z `'  _E  y )
2018, 19sylib 201 . . . . . . 7  |-  ( y  e.  om  ->  -.  A. z  e.  om  -.  z `'  _E  y
)
2120nrex 2853 . . . . . 6  |-  -.  E. y  e.  om  A. z  e.  om  -.  z `'  _E  y
22 ordom 6727 . . . . . . . 8  |-  Ord  om
23 eqid 2461 . . . . . . . . 9  |- OrdIso ( R ,  A )  = OrdIso
( R ,  A
)
2423oicl 8069 . . . . . . . 8  |-  Ord  dom OrdIso ( R ,  A )
25 ordtri1 5474 . . . . . . . 8  |-  ( ( Ord  om  /\  Ord  dom OrdIso ( R ,  A ) )  ->  ( om  C_ 
dom OrdIso ( R ,  A
)  <->  -.  dom OrdIso ( R ,  A )  e. 
om ) )
2622, 24, 25mp2an 683 . . . . . . 7  |-  ( om  C_  dom OrdIso ( R ,  A )  <->  -.  dom OrdIso ( R ,  A )  e. 
om )
27 wofib.1 . . . . . . . . . . 11  |-  A  e. 
_V
2823oion 8076 . . . . . . . . . . 11  |-  ( A  e.  _V  ->  dom OrdIso ( R ,  A )  e.  On )
2927, 28mp1i 13 . . . . . . . . . 10  |-  ( ( ( R  We  A  /\  `' R  We  A
)  /\  om  C_  dom OrdIso ( R ,  A ) )  ->  dom OrdIso ( R ,  A )  e.  On )
30 simpr 467 . . . . . . . . . 10  |-  ( ( ( R  We  A  /\  `' R  We  A
)  /\  om  C_  dom OrdIso ( R ,  A ) )  ->  om  C_  dom OrdIso ( R ,  A ) )
3129, 30ssexd 4563 . . . . . . . . 9  |-  ( ( ( R  We  A  /\  `' R  We  A
)  /\  om  C_  dom OrdIso ( R ,  A ) )  ->  om  e.  _V )
3223oiiso 8077 . . . . . . . . . . . . 13  |-  ( ( A  e.  _V  /\  R  We  A )  -> OrdIso ( R ,  A
)  Isom  _E  ,  R  ( dom OrdIso ( R ,  A ) ,  A
) )
3327, 32mpan 681 . . . . . . . . . . . 12  |-  ( R  We  A  -> OrdIso ( R ,  A )  Isom  _E  ,  R  ( dom OrdIso ( R ,  A ) ,  A ) )
34 isocnv2 6246 . . . . . . . . . . . 12  |-  (OrdIso ( R ,  A )  Isom  _E  ,  R  ( dom OrdIso ( R ,  A ) ,  A
)  <-> OrdIso ( R ,  A
)  Isom  `'  _E  ,  `' R ( dom OrdIso ( R ,  A ) ,  A ) )
3533, 34sylib 201 . . . . . . . . . . 11  |-  ( R  We  A  -> OrdIso ( R ,  A )  Isom  `'  _E  ,  `' R
( dom OrdIso ( R ,  A ) ,  A
) )
36 wefr 4842 . . . . . . . . . . 11  |-  ( `' R  We  A  ->  `' R  Fr  A
)
37 isofr 6257 . . . . . . . . . . . 12  |-  (OrdIso ( R ,  A )  Isom  `'  _E  ,  `' R
( dom OrdIso ( R ,  A ) ,  A
)  ->  ( `'  _E  Fr  dom OrdIso ( R ,  A )  <->  `' R  Fr  A ) )
3837biimpar 492 . . . . . . . . . . 11  |-  ( (OrdIso ( R ,  A
)  Isom  `'  _E  ,  `' R ( dom OrdIso ( R ,  A ) ,  A )  /\  `' R  Fr  A )  ->  `'  _E  Fr  dom OrdIso ( R ,  A ) )
3935, 36, 38syl2an 484 . . . . . . . . . 10  |-  ( ( R  We  A  /\  `' R  We  A
)  ->  `'  _E  Fr  dom OrdIso ( R ,  A ) )
4039adantr 471 . . . . . . . . 9  |-  ( ( ( R  We  A  /\  `' R  We  A
)  /\  om  C_  dom OrdIso ( R ,  A ) )  ->  `'  _E  Fr  dom OrdIso ( R ,  A ) )
41 1onn 7365 . . . . . . . . . 10  |-  1o  e.  om
42 ne0i 3748 . . . . . . . . . 10  |-  ( 1o  e.  om  ->  om  =/=  (/) )
4341, 42mp1i 13 . . . . . . . . 9  |-  ( ( ( R  We  A  /\  `' R  We  A
)  /\  om  C_  dom OrdIso ( R ,  A ) )  ->  om  =/=  (/) )
44 fri 4814 . . . . . . . . 9  |-  ( ( ( om  e.  _V  /\  `'  _E  Fr  dom OrdIso ( R ,  A ) )  /\  ( om  C_  dom OrdIso ( R ,  A )  /\  om  =/=  (/) ) )  ->  E. y  e.  om  A. z  e.  om  -.  z `'  _E  y
)
4531, 40, 30, 43, 44syl22anc 1277 . . . . . . . 8  |-  ( ( ( R  We  A  /\  `' R  We  A
)  /\  om  C_  dom OrdIso ( R ,  A ) )  ->  E. y  e.  om  A. z  e. 
om  -.  z `'  _E  y )
4645ex 440 . . . . . . 7  |-  ( ( R  We  A  /\  `' R  We  A
)  ->  ( om  C_ 
dom OrdIso ( R ,  A
)  ->  E. y  e.  om  A. z  e. 
om  -.  z `'  _E  y ) )
4726, 46syl5bir 226 . . . . . 6  |-  ( ( R  We  A  /\  `' R  We  A
)  ->  ( -.  dom OrdIso ( R ,  A
)  e.  om  ->  E. y  e.  om  A. z  e.  om  -.  z `'  _E  y ) )
4821, 47mt3i 131 . . . . 5  |-  ( ( R  We  A  /\  `' R  We  A
)  ->  dom OrdIso ( R ,  A )  e. 
om )
49 ssid 3462 . . . . 5  |-  dom OrdIso ( R ,  A )  C_  dom OrdIso ( R ,  A
)
50 ssnnfi 7816 . . . . 5  |-  ( ( dom OrdIso ( R ,  A )  e.  om  /\ 
dom OrdIso ( R ,  A
)  C_  dom OrdIso ( R ,  A ) )  ->  dom OrdIso ( R ,  A )  e.  Fin )
5148, 49, 50sylancl 673 . . . 4  |-  ( ( R  We  A  /\  `' R  We  A
)  ->  dom OrdIso ( R ,  A )  e. 
Fin )
52 simpl 463 . . . . . 6  |-  ( ( R  We  A  /\  `' R  We  A
)  ->  R  We  A )
5323oien 8078 . . . . . 6  |-  ( ( A  e.  _V  /\  R  We  A )  ->  dom OrdIso ( R ,  A )  ~~  A
)
5427, 52, 53sylancr 674 . . . . 5  |-  ( ( R  We  A  /\  `' R  We  A
)  ->  dom OrdIso ( R ,  A )  ~~  A )
55 enfi 7813 . . . . 5  |-  ( dom OrdIso ( R ,  A ) 
~~  A  ->  ( dom OrdIso ( R ,  A
)  e.  Fin  <->  A  e.  Fin ) )
5654, 55syl 17 . . . 4  |-  ( ( R  We  A  /\  `' R  We  A
)  ->  ( dom OrdIso ( R ,  A )  e.  Fin  <->  A  e.  Fin ) )
5751, 56mpbid 215 . . 3  |-  ( ( R  We  A  /\  `' R  We  A
)  ->  A  e.  Fin )
587, 57jca 539 . 2  |-  ( ( R  We  A  /\  `' R  We  A
)  ->  ( R  Or  A  /\  A  e. 
Fin ) )
595, 58impbii 192 1  |-  ( ( R  Or  A  /\  A  e.  Fin )  <->  ( R  We  A  /\  `' R  We  A
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 189    /\ wa 375    = wceq 1454    e. wcel 1897    =/= wne 2632   A.wral 2748   E.wrex 2749   _Vcvv 3056    C_ wss 3415   (/)c0 3742   class class class wbr 4415    _E cep 4761    Or wor 4772    Fr wfr 4808    We wwe 4810   `'ccnv 4851   dom cdm 4852   Ord word 5440   Oncon0 5441   suc csuc 5443    Isom wiso 5601   omcom 6718   1oc1o 7200    ~~ cen 7591   Fincfn 7594  OrdIsocoi 8049
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-8 1899  ax-9 1906  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441  ax-rep 4528  ax-sep 4538  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6609
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-eu 2313  df-mo 2314  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ne 2634  df-ral 2753  df-rex 2754  df-reu 2755  df-rmo 2756  df-rab 2757  df-v 3058  df-sbc 3279  df-csb 3375  df-dif 3418  df-un 3420  df-in 3422  df-ss 3429  df-pss 3431  df-nul 3743  df-if 3893  df-pw 3964  df-sn 3980  df-pr 3982  df-tp 3984  df-op 3986  df-uni 4212  df-iun 4293  df-br 4416  df-opab 4475  df-mpt 4476  df-tr 4511  df-eprel 4763  df-id 4767  df-po 4773  df-so 4774  df-fr 4811  df-se 4812  df-we 4813  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-pred 5398  df-ord 5444  df-on 5445  df-lim 5446  df-suc 5447  df-iota 5564  df-fun 5602  df-fn 5603  df-f 5604  df-f1 5605  df-fo 5606  df-f1o 5607  df-fv 5608  df-isom 5609  df-riota 6276  df-om 6719  df-wrecs 7053  df-recs 7115  df-1o 7207  df-er 7388  df-en 7595  df-fin 7598  df-oi 8050
This theorem is referenced by: (None)
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