MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  isfin4-3 Unicode version

Theorem isfin4-3 8028
Description: Alternate definition of IV-finite sets: they are strictly dominated by their successors. (Thus, the proper subset referred to in isfin4 8010 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 6570 . . . 4  |-  1o  e.  On
2 cdadom3 7901 . . . 4  |-  ( ( A  e. FinIV  /\  1o  e.  On )  ->  A  ~<_  ( A  +c  1o ) )
31, 2mpan2 652 . . 3  |-  ( A  e. FinIV  ->  A  ~<_  ( A  +c  1o ) )
4 0lt1o 6587 . . . . . . . . . 10  |-  (/)  e.  1o
51elexi 2873 . . . . . . . . . . 11  |-  1o  e.  _V
65snid 3743 . . . . . . . . . 10  |-  1o  e.  { 1o }
7 opelxpi 4800 . . . . . . . . . 10  |-  ( (
(/)  e.  1o  /\  1o  e.  { 1o } )  ->  <. (/) ,  1o >.  e.  ( 1o  X.  { 1o } ) )
84, 6, 7mp2an 653 . . . . . . . . 9  |-  <. (/) ,  1o >.  e.  ( 1o  X.  { 1o } )
9 elun2 3419 . . . . . . . . 9  |-  ( <. (/)
,  1o >.  e.  ( 1o  X.  { 1o } )  ->  <. (/) ,  1o >.  e.  ( ( A  X.  { (/) } )  u.  ( 1o  X.  { 1o } ) ) )
108, 9mp1i 11 . . . . . . . 8  |-  ( A 
~~  ( A  +c  1o )  ->  <. (/) ,  1o >.  e.  ( ( A  X.  { (/) } )  u.  ( 1o  X.  { 1o } ) ) )
11 relen 6953 . . . . . . . . . 10  |-  Rel  ~~
1211brrelexi 4808 . . . . . . . . 9  |-  ( A 
~~  ( A  +c  1o )  ->  A  e. 
_V )
13 cdaval 7883 . . . . . . . . 9  |-  ( ( A  e.  _V  /\  1o  e.  On )  -> 
( A  +c  1o )  =  ( ( A  X.  { (/) } )  u.  ( 1o  X.  { 1o } ) ) )
1412, 1, 13sylancl 643 . . . . . . . 8  |-  ( A 
~~  ( A  +c  1o )  ->  ( A  +c  1o )  =  ( ( A  X.  { (/) } )  u.  ( 1o  X.  { 1o } ) ) )
1510, 14eleqtrrd 2435 . . . . . . 7  |-  ( A 
~~  ( A  +c  1o )  ->  <. (/) ,  1o >.  e.  ( A  +c  1o ) )
16 1n0 6578 . . . . . . . 8  |-  1o  =/=  (/)
17 opelxp2 4802 . . . . . . . . . 10  |-  ( <. (/)
,  1o >.  e.  ( A  X.  { (/) } )  ->  1o  e.  {
(/) } )
18 elsni 3740 . . . . . . . . . 10  |-  ( 1o  e.  { (/) }  ->  1o  =  (/) )
1917, 18syl 15 . . . . . . . . 9  |-  ( <. (/)
,  1o >.  e.  ( A  X.  { (/) } )  ->  1o  =  (/) )
2019necon3ai 2561 . . . . . . . 8  |-  ( 1o  =/=  (/)  ->  -.  <. (/) ,  1o >.  e.  ( A  X.  { (/) } ) )
2116, 20mp1i 11 . . . . . . 7  |-  ( A 
~~  ( A  +c  1o )  ->  -.  <. (/)
,  1o >.  e.  ( A  X.  { (/) } ) )
22 ssun1 3414 . . . . . . . . 9  |-  ( A  X.  { (/) } ) 
C_  ( ( A  X.  { (/) } )  u.  ( 1o  X.  { 1o } ) )
2322, 14syl5sseqr 3303 . . . . . . . 8  |-  ( A 
~~  ( A  +c  1o )  ->  ( A  X.  { (/) } ) 
C_  ( A  +c  1o ) )
24 ssnelpss 3593 . . . . . . . 8  |-  ( ( A  X.  { (/) } )  C_  ( A  +c  1o )  ->  (
( <. (/) ,  1o >.  e.  ( A  +c  1o )  /\  -.  <. (/) ,  1o >.  e.  ( A  X.  { (/) } ) )  ->  ( A  X.  { (/) } )  C.  ( A  +c  1o ) ) )
2523, 24syl 15 . . . . . . 7  |-  ( A 
~~  ( A  +c  1o )  ->  ( (
<. (/) ,  1o >.  e.  ( A  +c  1o )  /\  -.  <. (/) ,  1o >.  e.  ( A  X.  { (/) } ) )  ->  ( A  X.  { (/) } )  C.  ( A  +c  1o ) ) )
2615, 21, 25mp2and 660 . . . . . 6  |-  ( A 
~~  ( A  +c  1o )  ->  ( A  X.  { (/) } ) 
C.  ( A  +c  1o ) )
27 0ex 4229 . . . . . . . 8  |-  (/)  e.  _V
28 xpsneng 7032 . . . . . . . 8  |-  ( ( A  e.  _V  /\  (/) 
e.  _V )  ->  ( A  X.  { (/) } ) 
~~  A )
2912, 27, 28sylancl 643 . . . . . . 7  |-  ( A 
~~  ( A  +c  1o )  ->  ( A  X.  { (/) } ) 
~~  A )
30 entr 6998 . . . . . . 7  |-  ( ( ( A  X.  { (/)
} )  ~~  A  /\  A  ~~  ( A  +c  1o ) )  ->  ( A  X.  { (/) } )  ~~  ( A  +c  1o ) )
3129, 30mpancom 650 . . . . . 6  |-  ( A 
~~  ( A  +c  1o )  ->  ( A  X.  { (/) } ) 
~~  ( A  +c  1o ) )
32 fin4i 8011 . . . . . 6  |-  ( ( ( A  X.  { (/)
} )  C.  ( A  +c  1o )  /\  ( A  X.  { (/) } )  ~~  ( A  +c  1o ) )  ->  -.  ( A  +c  1o )  e. FinIV )
3326, 31, 32syl2anc 642 . . . . 5  |-  ( A 
~~  ( A  +c  1o )  ->  -.  ( A  +c  1o )  e. FinIV )
34 fin4en1 8022 . . . . 5  |-  ( A 
~~  ( A  +c  1o )  ->  ( A  e. FinIV  ->  ( A  +c  1o )  e. FinIV ) )
3533, 34mtod 168 . . . 4  |-  ( A 
~~  ( A  +c  1o )  ->  -.  A  e. FinIV
)
3635con2i 112 . . 3  |-  ( A  e. FinIV  ->  -.  A  ~~  ( A  +c  1o ) )
37 brsdom 6969 . . 3  |-  ( A 
~<  ( A  +c  1o ) 
<->  ( A  ~<_  ( A  +c  1o )  /\  -.  A  ~~  ( A  +c  1o ) ) )
383, 36, 37sylanbrc 645 . 2  |-  ( A  e. FinIV  ->  A  ~<  ( A  +c  1o ) )
39 sdomnen 6975 . . . 4  |-  ( A 
~<  ( A  +c  1o )  ->  -.  A  ~~  ( A  +c  1o ) )
40 infcda1 7906 . . . . 5  |-  ( om  ~<_  A  ->  ( A  +c  1o )  ~~  A
)
41 ensym 6995 . . . . 5  |-  ( ( A  +c  1o ) 
~~  A  ->  A  ~~  ( A  +c  1o ) )
4240, 41syl 15 . . . 4  |-  ( om  ~<_  A  ->  A  ~~  ( A  +c  1o ) )
4339, 42nsyl 113 . . 3  |-  ( A 
~<  ( A  +c  1o )  ->  -.  om  ~<_  A )
44 relsdom 6955 . . . . 5  |-  Rel  ~<
4544brrelexi 4808 . . . 4  |-  ( A 
~<  ( A  +c  1o )  ->  A  e.  _V )
46 isfin4-2 8027 . . . 4  |-  ( A  e.  _V  ->  ( A  e. FinIV 
<->  -.  om  ~<_  A ) )
4745, 46syl 15 . . 3  |-  ( A 
~<  ( A  +c  1o )  ->  ( A  e. FinIV  <->  -.  om  ~<_  A ) )
4843, 47mpbird 223 . 2  |-  ( A 
~<  ( A  +c  1o )  ->  A  e. FinIV )
4938, 48impbii 180 1  |-  ( A  e. FinIV  <-> 
A  ~<  ( A  +c  1o ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1642    e. wcel 1710    =/= wne 2521   _Vcvv 2864    u. cun 3226    C_ wss 3228    C. wpss 3229   (/)c0 3531   {csn 3716   <.cop 3719   class class class wbr 4102   Oncon0 4471   omcom 4735    X. cxp 4766  (class class class)co 5942   1oc1o 6556    ~~ cen 6945    ~<_ cdom 6946    ~< csdm 6947    +c ccda 7880  FinIVcfin4 7993
This theorem is referenced by:  fin45  8105  finngch  8364  gchinf  8366
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4210  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-reu 2626  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3907  df-int 3942  df-iun 3986  df-br 4103  df-opab 4157  df-mpt 4158  df-tr 4193  df-eprel 4384  df-id 4388  df-po 4393  df-so 4394  df-fr 4431  df-we 4433  df-ord 4474  df-on 4475  df-lim 4476  df-suc 4477  df-om 4736  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-f1 5339  df-fo 5340  df-f1o 5341  df-fv 5342  df-ov 5945  df-oprab 5946  df-mpt2 5947  df-recs 6472  df-rdg 6507  df-1o 6563  df-er 6744  df-en 6949  df-dom 6950  df-sdom 6951  df-fin 6952  df-cda 7881  df-fin4 8000
  Copyright terms: Public domain W3C validator