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Theorem isfin4-3 8226
Description: Alternate definition of IV-finite sets: they are strictly dominated by their successors. (Thus, the proper subset referred to in isfin4 8208 can be assumed to be only a singleton smaller than the original.) (Contributed by Mario Carneiro, 18-May-2015.)
Assertion
Ref Expression
isfin4-3  |-  ( A  e. FinIV  <-> 
A  ~<  ( A  +c  1o ) )

Proof of Theorem isfin4-3
StepHypRef Expression
1 1on 6760 . . . 4  |-  1o  e.  On
2 cdadom3 8099 . . . 4  |-  ( ( A  e. FinIV  /\  1o  e.  On )  ->  A  ~<_  ( A  +c  1o ) )
31, 2mpan2 654 . . 3  |-  ( A  e. FinIV  ->  A  ~<_  ( A  +c  1o ) )
4 ssun1 3496 . . . . . . . 8  |-  ( A  X.  { (/) } ) 
C_  ( ( A  X.  { (/) } )  u.  ( 1o  X.  { 1o } ) )
5 relen 7143 . . . . . . . . . 10  |-  Rel  ~~
65brrelexi 4947 . . . . . . . . 9  |-  ( A 
~~  ( A  +c  1o )  ->  A  e. 
_V )
7 cdaval 8081 . . . . . . . . 9  |-  ( ( A  e.  _V  /\  1o  e.  On )  -> 
( A  +c  1o )  =  ( ( A  X.  { (/) } )  u.  ( 1o  X.  { 1o } ) ) )
86, 1, 7sylancl 645 . . . . . . . 8  |-  ( A 
~~  ( A  +c  1o )  ->  ( A  +c  1o )  =  ( ( A  X.  { (/) } )  u.  ( 1o  X.  { 1o } ) ) )
94, 8syl5sseqr 3383 . . . . . . 7  |-  ( A 
~~  ( A  +c  1o )  ->  ( A  X.  { (/) } ) 
C_  ( A  +c  1o ) )
10 0lt1o 6777 . . . . . . . . . 10  |-  (/)  e.  1o
111elexi 2971 . . . . . . . . . . 11  |-  1o  e.  _V
1211snid 3865 . . . . . . . . . 10  |-  1o  e.  { 1o }
13 opelxpi 4939 . . . . . . . . . 10  |-  ( (
(/)  e.  1o  /\  1o  e.  { 1o } )  ->  <. (/) ,  1o >.  e.  ( 1o  X.  { 1o } ) )
1410, 12, 13mp2an 655 . . . . . . . . 9  |-  <. (/) ,  1o >.  e.  ( 1o  X.  { 1o } )
15 elun2 3501 . . . . . . . . 9  |-  ( <. (/)
,  1o >.  e.  ( 1o  X.  { 1o } )  ->  <. (/) ,  1o >.  e.  ( ( A  X.  { (/) } )  u.  ( 1o  X.  { 1o } ) ) )
1614, 15mp1i 12 . . . . . . . 8  |-  ( A 
~~  ( A  +c  1o )  ->  <. (/) ,  1o >.  e.  ( ( A  X.  { (/) } )  u.  ( 1o  X.  { 1o } ) ) )
1716, 8eleqtrrd 2519 . . . . . . 7  |-  ( A 
~~  ( A  +c  1o )  ->  <. (/) ,  1o >.  e.  ( A  +c  1o ) )
18 1n0 6768 . . . . . . . 8  |-  1o  =/=  (/)
19 opelxp2 4941 . . . . . . . . . 10  |-  ( <. (/)
,  1o >.  e.  ( A  X.  { (/) } )  ->  1o  e.  {
(/) } )
20 elsni 3862 . . . . . . . . . 10  |-  ( 1o  e.  { (/) }  ->  1o  =  (/) )
2119, 20syl 16 . . . . . . . . 9  |-  ( <. (/)
,  1o >.  e.  ( A  X.  { (/) } )  ->  1o  =  (/) )
2221necon3ai 2650 . . . . . . . 8  |-  ( 1o  =/=  (/)  ->  -.  <. (/) ,  1o >.  e.  ( A  X.  { (/) } ) )
2318, 22mp1i 12 . . . . . . 7  |-  ( A 
~~  ( A  +c  1o )  ->  -.  <. (/)
,  1o >.  e.  ( A  X.  { (/) } ) )
249, 17, 23ssnelpssd 3716 . . . . . 6  |-  ( A 
~~  ( A  +c  1o )  ->  ( A  X.  { (/) } ) 
C.  ( A  +c  1o ) )
25 0ex 4364 . . . . . . . 8  |-  (/)  e.  _V
26 xpsneng 7222 . . . . . . . 8  |-  ( ( A  e.  _V  /\  (/) 
e.  _V )  ->  ( A  X.  { (/) } ) 
~~  A )
276, 25, 26sylancl 645 . . . . . . 7  |-  ( A 
~~  ( A  +c  1o )  ->  ( A  X.  { (/) } ) 
~~  A )
28 entr 7188 . . . . . . 7  |-  ( ( ( A  X.  { (/)
} )  ~~  A  /\  A  ~~  ( A  +c  1o ) )  ->  ( A  X.  { (/) } )  ~~  ( A  +c  1o ) )
2927, 28mpancom 652 . . . . . 6  |-  ( A 
~~  ( A  +c  1o )  ->  ( A  X.  { (/) } ) 
~~  ( A  +c  1o ) )
30 fin4i 8209 . . . . . 6  |-  ( ( ( A  X.  { (/)
} )  C.  ( A  +c  1o )  /\  ( A  X.  { (/) } )  ~~  ( A  +c  1o ) )  ->  -.  ( A  +c  1o )  e. FinIV )
3124, 29, 30syl2anc 644 . . . . 5  |-  ( A 
~~  ( A  +c  1o )  ->  -.  ( A  +c  1o )  e. FinIV )
32 fin4en1 8220 . . . . 5  |-  ( A 
~~  ( A  +c  1o )  ->  ( A  e. FinIV  ->  ( A  +c  1o )  e. FinIV ) )
3331, 32mtod 171 . . . 4  |-  ( A 
~~  ( A  +c  1o )  ->  -.  A  e. FinIV
)
3433con2i 115 . . 3  |-  ( A  e. FinIV  ->  -.  A  ~~  ( A  +c  1o ) )
35 brsdom 7159 . . 3  |-  ( A 
~<  ( A  +c  1o ) 
<->  ( A  ~<_  ( A  +c  1o )  /\  -.  A  ~~  ( A  +c  1o ) ) )
363, 34, 35sylanbrc 647 . 2  |-  ( A  e. FinIV  ->  A  ~<  ( A  +c  1o ) )
37 sdomnen 7165 . . . 4  |-  ( A 
~<  ( A  +c  1o )  ->  -.  A  ~~  ( A  +c  1o ) )
38 infcda1 8104 . . . . 5  |-  ( om  ~<_  A  ->  ( A  +c  1o )  ~~  A
)
3938ensymd 7187 . . . 4  |-  ( om  ~<_  A  ->  A  ~~  ( A  +c  1o ) )
4037, 39nsyl 116 . . 3  |-  ( A 
~<  ( A  +c  1o )  ->  -.  om  ~<_  A )
41 relsdom 7145 . . . . 5  |-  Rel  ~<
4241brrelexi 4947 . . . 4  |-  ( A 
~<  ( A  +c  1o )  ->  A  e.  _V )
43 isfin4-2 8225 . . . 4  |-  ( A  e.  _V  ->  ( A  e. FinIV 
<->  -.  om  ~<_  A ) )
4442, 43syl 16 . . 3  |-  ( A 
~<  ( A  +c  1o )  ->  ( A  e. FinIV  <->  -.  om  ~<_  A ) )
4540, 44mpbird 225 . 2  |-  ( A 
~<  ( A  +c  1o )  ->  A  e. FinIV )
4636, 45impbii 182 1  |-  ( A  e. FinIV  <-> 
A  ~<  ( A  +c  1o ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 178    = wceq 1653    e. wcel 1727    =/= wne 2605   _Vcvv 2962    u. cun 3304    C. wpss 3307   (/)c0 3613   {csn 3838   <.cop 3841   class class class wbr 4237   Oncon0 4610   omcom 4874    X. cxp 4905  (class class class)co 6110   1oc1o 6746    ~~ cen 7135    ~<_ cdom 7136    ~< csdm 7137    +c ccda 8078  FinIVcfin4 8191
This theorem is referenced by:  fin45  8303  finngch  8561  gchinf  8563
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-rep 4345  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432  ax-un 4730
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2716  df-rex 2717  df-reu 2718  df-rab 2720  df-v 2964  df-sbc 3168  df-csb 3268  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-pss 3322  df-nul 3614  df-if 3764  df-pw 3825  df-sn 3844  df-pr 3845  df-tp 3846  df-op 3847  df-uni 4040  df-int 4075  df-iun 4119  df-br 4238  df-opab 4292  df-mpt 4293  df-tr 4328  df-eprel 4523  df-id 4527  df-po 4532  df-so 4533  df-fr 4570  df-we 4572  df-ord 4613  df-on 4614  df-lim 4615  df-suc 4616  df-om 4875  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-ima 4920  df-iota 5447  df-fun 5485  df-fn 5486  df-f 5487  df-f1 5488  df-fo 5489  df-f1o 5490  df-fv 5491  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-recs 6662  df-rdg 6697  df-1o 6753  df-er 6934  df-en 7139  df-dom 7140  df-sdom 7141  df-fin 7142  df-cda 8079  df-fin4 8198
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