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Theorem wofib 7764
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 7566 . . 3  |-  ( ( R  Or  A  /\  A  e.  Fin )  ->  R  We  A )
2 cnvso 5381 . . . 4  |-  ( R  Or  A  <->  `' R  Or  A )
3 wofi 7566 . . . 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 4716 . . . 4  |-  ( R  We  A  ->  R  Or  A )
76adantr 465 . . 3  |-  ( ( R  We  A  /\  `' R  We  A
)  ->  R  Or  A )
8 peano2 6501 . . . . . . . . 9  |-  ( y  e.  om  ->  suc  y  e.  om )
9 sucidg 4802 . . . . . . . . 9  |-  ( y  e.  om  ->  y  e.  suc  y )
10 vex 2980 . . . . . . . . . . . . 13  |-  z  e. 
_V
11 vex 2980 . . . . . . . . . . . . 13  |-  y  e. 
_V
1210, 11brcnv 5027 . . . . . . . . . . . 12  |-  ( z `'  _E  y  <->  y  _E  z )
13 epel 4640 . . . . . . . . . . . 12  |-  ( y  _E  z  <->  y  e.  z )
1412, 13bitri 249 . . . . . . . . . . 11  |-  ( z `'  _E  y  <->  y  e.  z )
15 eleq2 2504 . . . . . . . . . . 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 3078 . . . . . . . . 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 2733 . . . . . . . 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 2823 . . . . . 6  |-  -.  E. y  e.  om  A. z  e.  om  -.  z `'  _E  y
22 ordom 6490 . . . . . . . 8  |-  Ord  om
23 eqid 2443 . . . . . . . . 9  |- OrdIso ( R ,  A )  = OrdIso
( R ,  A
)
2423oicl 7748 . . . . . . . 8  |-  Ord  dom OrdIso ( R ,  A )
25 ordtri1 4757 . . . . . . . 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 7755 . . . . . . . . . . 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 4444 . . . . . . . . 9  |-  ( ( ( R  We  A  /\  `' R  We  A
)  /\  om  C_  dom OrdIso ( R ,  A ) )  ->  om  e.  _V )
3223oiiso 7756 . . . . . . . . . . . . 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 6027 . . . . . . . . . . . 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 4715 . . . . . . . . . . 11  |-  ( `' R  We  A  ->  `' R  Fr  A
)
37 isofr 6038 . . . . . . . . . . . 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 7083 . . . . . . . . . 10  |-  1o  e.  om
42 ne0i 3648 . . . . . . . . . 10  |-  ( 1o  e.  om  ->  om  =/=  (/) )
4341, 42mp1i 12 . . . . . . . . 9  |-  ( ( ( R  We  A  /\  `' R  We  A
)  /\  om  C_  dom OrdIso ( R ,  A ) )  ->  om  =/=  (/) )
44 fri 4687 . . . . . . . . 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 1219 . . . . . . . 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 3380 . . . . 5  |-  dom OrdIso ( R ,  A )  C_  dom OrdIso ( R ,  A
)
50 ssnnfi 7537 . . . . 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 7757 . . . . . 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 7534 . . . . 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 1369    e. wcel 1756    =/= wne 2611   A.wral 2720   E.wrex 2721   _Vcvv 2977    C_ wss 3333   (/)c0 3642   class class class wbr 4297    _E cep 4635    Or wor 4645    Fr wfr 4681    We wwe 4683   Ord word 4723   Oncon0 4724   suc csuc 4726   `'ccnv 4844   dom cdm 4845    Isom wiso 5424   omcom 6481   1oc1o 6918    ~~ cen 7312   Fincfn 7315  OrdIsocoi 7728
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-se 4685  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  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-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-isom 5432  df-riota 6057  df-om 6482  df-recs 6837  df-1o 6925  df-er 7106  df-en 7316  df-fin 7319  df-oi 7729
This theorem is referenced by: (None)
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