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Theorem wofib 7970
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 7769 . . 3  |-  ( ( R  Or  A  /\  A  e.  Fin )  ->  R  We  A )
2 cnvso 5546 . . . 4  |-  ( R  Or  A  <->  `' R  Or  A )
3 wofi 7769 . . . 4  |-  ( ( `' R  Or  A  /\  A  e.  Fin )  ->  `' R  We  A )
42, 3sylanb 472 . . 3  |-  ( ( R  Or  A  /\  A  e.  Fin )  ->  `' R  We  A
)
51, 4jca 532 . 2  |-  ( ( R  Or  A  /\  A  e.  Fin )  ->  ( R  We  A  /\  `' R  We  A
) )
6 weso 4870 . . . 4  |-  ( R  We  A  ->  R  Or  A )
76adantr 465 . . 3  |-  ( ( R  We  A  /\  `' R  We  A
)  ->  R  Or  A )
8 peano2 6704 . . . . . . . . 9  |-  ( y  e.  om  ->  suc  y  e.  om )
9 sucidg 4956 . . . . . . . . 9  |-  ( y  e.  om  ->  y  e.  suc  y )
10 vex 3116 . . . . . . . . . . . . 13  |-  z  e. 
_V
11 vex 3116 . . . . . . . . . . . . 13  |-  y  e. 
_V
1210, 11brcnv 5185 . . . . . . . . . . . 12  |-  ( z `'  _E  y  <->  y  _E  z )
13 epel 4794 . . . . . . . . . . . 12  |-  ( y  _E  z  <->  y  e.  z )
1412, 13bitri 249 . . . . . . . . . . 11  |-  ( z `'  _E  y  <->  y  e.  z )
15 eleq2 2540 . . . . . . . . . . 11  |-  ( z  =  suc  y  -> 
( y  e.  z  <-> 
y  e.  suc  y
) )
1614, 15syl5bb 257 . . . . . . . . . 10  |-  ( z  =  suc  y  -> 
( z `'  _E  y 
<->  y  e.  suc  y
) )
1716rspcev 3214 . . . . . . . . 9  |-  ( ( suc  y  e.  om  /\  y  e.  suc  y
)  ->  E. z  e.  om  z `'  _E  y )
188, 9, 17syl2anc 661 . . . . . . . 8  |-  ( y  e.  om  ->  E. z  e.  om  z `'  _E  y )
19 dfrex2 2915 . . . . . . . 8  |-  ( E. z  e.  om  z `'  _E  y  <->  -.  A. z  e.  om  -.  z `'  _E  y )
2018, 19sylib 196 . . . . . . 7  |-  ( y  e.  om  ->  -.  A. z  e.  om  -.  z `'  _E  y
)
2120nrex 2919 . . . . . 6  |-  -.  E. y  e.  om  A. z  e.  om  -.  z `'  _E  y
22 ordom 6693 . . . . . . . 8  |-  Ord  om
23 eqid 2467 . . . . . . . . 9  |- OrdIso ( R ,  A )  = OrdIso
( R ,  A
)
2423oicl 7954 . . . . . . . 8  |-  Ord  dom OrdIso ( R ,  A )
25 ordtri1 4911 . . . . . . . 8  |-  ( ( Ord  om  /\  Ord  dom OrdIso ( R ,  A ) )  ->  ( om  C_ 
dom OrdIso ( R ,  A
)  <->  -.  dom OrdIso ( R ,  A )  e. 
om ) )
2622, 24, 25mp2an 672 . . . . . . 7  |-  ( om  C_  dom OrdIso ( R ,  A )  <->  -.  dom OrdIso ( R ,  A )  e. 
om )
27 wofib.1 . . . . . . . . . . 11  |-  A  e. 
_V
2823oion 7961 . . . . . . . . . . 11  |-  ( A  e.  _V  ->  dom OrdIso ( R ,  A )  e.  On )
2927, 28mp1i 12 . . . . . . . . . 10  |-  ( ( ( R  We  A  /\  `' R  We  A
)  /\  om  C_  dom OrdIso ( R ,  A ) )  ->  dom OrdIso ( R ,  A )  e.  On )
30 simpr 461 . . . . . . . . . 10  |-  ( ( ( R  We  A  /\  `' R  We  A
)  /\  om  C_  dom OrdIso ( R ,  A ) )  ->  om  C_  dom OrdIso ( R ,  A ) )
3129, 30ssexd 4594 . . . . . . . . 9  |-  ( ( ( R  We  A  /\  `' R  We  A
)  /\  om  C_  dom OrdIso ( R ,  A ) )  ->  om  e.  _V )
3223oiiso 7962 . . . . . . . . . . . . 13  |-  ( ( A  e.  _V  /\  R  We  A )  -> OrdIso ( R ,  A
)  Isom  _E  ,  R  ( dom OrdIso ( R ,  A ) ,  A
) )
3327, 32mpan 670 . . . . . . . . . . . 12  |-  ( R  We  A  -> OrdIso ( R ,  A )  Isom  _E  ,  R  ( dom OrdIso ( R ,  A ) ,  A ) )
34 isocnv2 6215 . . . . . . . . . . . 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 196 . . . . . . . . . . 11  |-  ( R  We  A  -> OrdIso ( R ,  A )  Isom  `'  _E  ,  `' R
( dom OrdIso ( R ,  A ) ,  A
) )
36 wefr 4869 . . . . . . . . . . 11  |-  ( `' R  We  A  ->  `' R  Fr  A
)
37 isofr 6226 . . . . . . . . . . . 12  |-  (OrdIso ( R ,  A )  Isom  `'  _E  ,  `' R
( dom OrdIso ( R ,  A ) ,  A
)  ->  ( `'  _E  Fr  dom OrdIso ( R ,  A )  <->  `' R  Fr  A ) )
3837biimpar 485 . . . . . . . . . . 11  |-  ( (OrdIso ( R ,  A
)  Isom  `'  _E  ,  `' R ( dom OrdIso ( R ,  A ) ,  A )  /\  `' R  Fr  A )  ->  `'  _E  Fr  dom OrdIso ( R ,  A ) )
3935, 36, 38syl2an 477 . . . . . . . . . 10  |-  ( ( R  We  A  /\  `' R  We  A
)  ->  `'  _E  Fr  dom OrdIso ( R ,  A ) )
4039adantr 465 . . . . . . . . 9  |-  ( ( ( R  We  A  /\  `' R  We  A
)  /\  om  C_  dom OrdIso ( R ,  A ) )  ->  `'  _E  Fr  dom OrdIso ( R ,  A ) )
41 1onn 7288 . . . . . . . . . 10  |-  1o  e.  om
42 ne0i 3791 . . . . . . . . . 10  |-  ( 1o  e.  om  ->  om  =/=  (/) )
4341, 42mp1i 12 . . . . . . . . 9  |-  ( ( ( R  We  A  /\  `' R  We  A
)  /\  om  C_  dom OrdIso ( R ,  A ) )  ->  om  =/=  (/) )
44 fri 4841 . . . . . . . . 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 1229 . . . . . . . 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 434 . . . . . . 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 218 . . . . . 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 126 . . . . 5  |-  ( ( R  We  A  /\  `' R  We  A
)  ->  dom OrdIso ( R ,  A )  e. 
om )
49 ssid 3523 . . . . 5  |-  dom OrdIso ( R ,  A )  C_  dom OrdIso ( R ,  A
)
50 ssnnfi 7739 . . . . 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 662 . . . 4  |-  ( ( R  We  A  /\  `' R  We  A
)  ->  dom OrdIso ( R ,  A )  e. 
Fin )
52 simpl 457 . . . . . 6  |-  ( ( R  We  A  /\  `' R  We  A
)  ->  R  We  A )
5323oien 7963 . . . . . 6  |-  ( ( A  e.  _V  /\  R  We  A )  ->  dom OrdIso ( R ,  A )  ~~  A
)
5427, 52, 53sylancr 663 . . . . 5  |-  ( ( R  We  A  /\  `' R  We  A
)  ->  dom OrdIso ( R ,  A )  ~~  A )
55 enfi 7736 . . . . 5  |-  ( dom OrdIso ( R ,  A ) 
~~  A  ->  ( dom OrdIso ( R ,  A
)  e.  Fin  <->  A  e.  Fin ) )
5654, 55syl 16 . . . 4  |-  ( ( R  We  A  /\  `' R  We  A
)  ->  ( dom OrdIso ( R ,  A )  e.  Fin  <->  A  e.  Fin ) )
5751, 56mpbid 210 . . 3  |-  ( ( R  We  A  /\  `' R  We  A
)  ->  A  e.  Fin )
587, 57jca 532 . 2  |-  ( ( R  We  A  /\  `' R  We  A
)  ->  ( R  Or  A  /\  A  e. 
Fin ) )
595, 58impbii 188 1  |-  ( ( R  Or  A  /\  A  e.  Fin )  <->  ( R  We  A  /\  `' R  We  A
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2814   E.wrex 2815   _Vcvv 3113    C_ wss 3476   (/)c0 3785   class class class wbr 4447    _E cep 4789    Or wor 4799    Fr wfr 4835    We wwe 4837   Ord word 4877   Oncon0 4878   suc csuc 4880   `'ccnv 4998   dom cdm 4999    Isom wiso 5589   omcom 6684   1oc1o 7123    ~~ cen 7513   Fincfn 7516  OrdIsocoi 7934
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 6576
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-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 6245  df-om 6685  df-recs 7042  df-1o 7130  df-er 7311  df-en 7517  df-fin 7520  df-oi 7935
This theorem is referenced by: (None)
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