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Theorem isfin4-3 7941
Description: Alternate definition of IV-finite sets: they are strictly dominated by their successors. (Thus the proper subset referred to in isfin4 7923 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 6486 . . . 4  |-  1o  e.  On
2 cdadom3 7814 . . . 4  |-  ( ( A  e. FinIV  /\  1o  e.  On )  ->  A  ~<_  ( A  +c  1o ) )
31, 2mpan2 652 . . 3  |-  ( A  e. FinIV  ->  A  ~<_  ( A  +c  1o ) )
4 0lt1o 6503 . . . . . . . . . 10  |-  (/)  e.  1o
51elexi 2797 . . . . . . . . . . 11  |-  1o  e.  _V
65snid 3667 . . . . . . . . . 10  |-  1o  e.  { 1o }
7 opelxpi 4721 . . . . . . . . . 10  |-  ( (
(/)  e.  1o  /\  1o  e.  { 1o } )  ->  <. (/) ,  1o >.  e.  ( 1o  X.  { 1o } ) )
84, 6, 7mp2an 653 . . . . . . . . 9  |-  <. (/) ,  1o >.  e.  ( 1o  X.  { 1o } )
9 elun2 3343 . . . . . . . . 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 6868 . . . . . . . . . 10  |-  Rel  ~~
1211brrelexi 4729 . . . . . . . . 9  |-  ( A 
~~  ( A  +c  1o )  ->  A  e. 
_V )
13 cdaval 7796 . . . . . . . . 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 2360 . . . . . . 7  |-  ( A 
~~  ( A  +c  1o )  ->  <. (/) ,  1o >.  e.  ( A  +c  1o ) )
16 1n0 6494 . . . . . . . 8  |-  1o  =/=  (/)
17 opelxp2 4723 . . . . . . . . . 10  |-  ( <. (/)
,  1o >.  e.  ( A  X.  { (/) } )  ->  1o  e.  {
(/) } )
18 elsni 3664 . . . . . . . . . 10  |-  ( 1o  e.  { (/) }  ->  1o  =  (/) )
1917, 18syl 15 . . . . . . . . 9  |-  ( <. (/)
,  1o >.  e.  ( A  X.  { (/) } )  ->  1o  =  (/) )
2019necon3ai 2486 . . . . . . . 8  |-  ( 1o  =/=  (/)  ->  -.  <. (/) ,  1o >.  e.  ( A  X.  { (/) } ) )
2116, 20mp1i 11 . . . . . . 7  |-  ( A 
~~  ( A  +c  1o )  ->  -.  <. (/)
,  1o >.  e.  ( A  X.  { (/) } ) )
22 ssun1 3338 . . . . . . . . 9  |-  ( A  X.  { (/) } ) 
C_  ( ( A  X.  { (/) } )  u.  ( 1o  X.  { 1o } ) )
2322, 14syl5sseqr 3227 . . . . . . . 8  |-  ( A 
~~  ( A  +c  1o )  ->  ( A  X.  { (/) } ) 
C_  ( A  +c  1o ) )
24 ssnelpss 3517 . . . . . . . 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 4150 . . . . . . . 8  |-  (/)  e.  _V
28 xpsneng 6947 . . . . . . . 8  |-  ( ( A  e.  _V  /\  (/) 
e.  _V )  ->  ( A  X.  { (/) } ) 
~~  A )
2912, 27, 28sylancl 643 . . . . . . 7  |-  ( A 
~~  ( A  +c  1o )  ->  ( A  X.  { (/) } ) 
~~  A )
30 entr 6913 . . . . . . 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 7924 . . . . . 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 7935 . . . . 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 6884 . . 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 6890 . . . 4  |-  ( A 
~<  ( A  +c  1o )  ->  -.  A  ~~  ( A  +c  1o ) )
40 infcda1 7819 . . . . 5  |-  ( om  ~<_  A  ->  ( A  +c  1o )  ~~  A
)
41 ensym 6910 . . . . 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 6870 . . . . 5  |-  Rel  ~<
4544brrelexi 4729 . . . 4  |-  ( A 
~<  ( A  +c  1o )  ->  A  e.  _V )
46 isfin4-2 7940 . . . 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 1623    e. wcel 1684    =/= wne 2446   _Vcvv 2788    u. cun 3150    C_ wss 3152    C. wpss 3153   (/)c0 3455   {csn 3640   <.cop 3643   class class class wbr 4023   Oncon0 4392   omcom 4656    X. cxp 4687  (class class class)co 5858   1oc1o 6472    ~~ cen 6860    ~<_ cdom 6861    ~< csdm 6862    +c ccda 7793  FinIVcfin4 7906
This theorem is referenced by:  fin45  8018  finngch  8277  gchinf  8279
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-recs 6388  df-rdg 6423  df-1o 6479  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-cda 7794  df-fin4 7913
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