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Theorem isfin4-3 8151
Description: Alternate definition of IV-finite sets: they are strictly dominated by their successors. (Thus, the proper subset referred to in isfin4 8133 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 6690 . . . 4  |-  1o  e.  On
2 cdadom3 8024 . . . 4  |-  ( ( A  e. FinIV  /\  1o  e.  On )  ->  A  ~<_  ( A  +c  1o ) )
31, 2mpan2 653 . . 3  |-  ( A  e. FinIV  ->  A  ~<_  ( A  +c  1o ) )
4 ssun1 3470 . . . . . . . 8  |-  ( A  X.  { (/) } ) 
C_  ( ( A  X.  { (/) } )  u.  ( 1o  X.  { 1o } ) )
5 relen 7073 . . . . . . . . . 10  |-  Rel  ~~
65brrelexi 4877 . . . . . . . . 9  |-  ( A 
~~  ( A  +c  1o )  ->  A  e. 
_V )
7 cdaval 8006 . . . . . . . . 9  |-  ( ( A  e.  _V  /\  1o  e.  On )  -> 
( A  +c  1o )  =  ( ( A  X.  { (/) } )  u.  ( 1o  X.  { 1o } ) ) )
86, 1, 7sylancl 644 . . . . . . . 8  |-  ( A 
~~  ( A  +c  1o )  ->  ( A  +c  1o )  =  ( ( A  X.  { (/) } )  u.  ( 1o  X.  { 1o } ) ) )
94, 8syl5sseqr 3357 . . . . . . 7  |-  ( A 
~~  ( A  +c  1o )  ->  ( A  X.  { (/) } ) 
C_  ( A  +c  1o ) )
10 0lt1o 6707 . . . . . . . . . 10  |-  (/)  e.  1o
111elexi 2925 . . . . . . . . . . 11  |-  1o  e.  _V
1211snid 3801 . . . . . . . . . 10  |-  1o  e.  { 1o }
13 opelxpi 4869 . . . . . . . . . 10  |-  ( (
(/)  e.  1o  /\  1o  e.  { 1o } )  ->  <. (/) ,  1o >.  e.  ( 1o  X.  { 1o } ) )
1410, 12, 13mp2an 654 . . . . . . . . 9  |-  <. (/) ,  1o >.  e.  ( 1o  X.  { 1o } )
15 elun2 3475 . . . . . . . . 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 2481 . . . . . . 7  |-  ( A 
~~  ( A  +c  1o )  ->  <. (/) ,  1o >.  e.  ( A  +c  1o ) )
18 1n0 6698 . . . . . . . 8  |-  1o  =/=  (/)
19 opelxp2 4871 . . . . . . . . . 10  |-  ( <. (/)
,  1o >.  e.  ( A  X.  { (/) } )  ->  1o  e.  {
(/) } )
20 elsni 3798 . . . . . . . . . 10  |-  ( 1o  e.  { (/) }  ->  1o  =  (/) )
2119, 20syl 16 . . . . . . . . 9  |-  ( <. (/)
,  1o >.  e.  ( A  X.  { (/) } )  ->  1o  =  (/) )
2221necon3ai 2607 . . . . . . . 8  |-  ( 1o  =/=  (/)  ->  -.  <. (/) ,  1o >.  e.  ( A  X.  { (/) } ) )
2318, 22mp1i 12 . . . . . . 7  |-  ( A 
~~  ( A  +c  1o )  ->  -.  <. (/)
,  1o >.  e.  ( A  X.  { (/) } ) )
249, 17, 23ssnelpssd 3652 . . . . . 6  |-  ( A 
~~  ( A  +c  1o )  ->  ( A  X.  { (/) } ) 
C.  ( A  +c  1o ) )
25 0ex 4299 . . . . . . . 8  |-  (/)  e.  _V
26 xpsneng 7152 . . . . . . . 8  |-  ( ( A  e.  _V  /\  (/) 
e.  _V )  ->  ( A  X.  { (/) } ) 
~~  A )
276, 25, 26sylancl 644 . . . . . . 7  |-  ( A 
~~  ( A  +c  1o )  ->  ( A  X.  { (/) } ) 
~~  A )
28 entr 7118 . . . . . . 7  |-  ( ( ( A  X.  { (/)
} )  ~~  A  /\  A  ~~  ( A  +c  1o ) )  ->  ( A  X.  { (/) } )  ~~  ( A  +c  1o ) )
2927, 28mpancom 651 . . . . . 6  |-  ( A 
~~  ( A  +c  1o )  ->  ( A  X.  { (/) } ) 
~~  ( A  +c  1o ) )
30 fin4i 8134 . . . . . 6  |-  ( ( ( A  X.  { (/)
} )  C.  ( A  +c  1o )  /\  ( A  X.  { (/) } )  ~~  ( A  +c  1o ) )  ->  -.  ( A  +c  1o )  e. FinIV )
3124, 29, 30syl2anc 643 . . . . 5  |-  ( A 
~~  ( A  +c  1o )  ->  -.  ( A  +c  1o )  e. FinIV )
32 fin4en1 8145 . . . . 5  |-  ( A 
~~  ( A  +c  1o )  ->  ( A  e. FinIV  ->  ( A  +c  1o )  e. FinIV ) )
3331, 32mtod 170 . . . 4  |-  ( A 
~~  ( A  +c  1o )  ->  -.  A  e. FinIV
)
3433con2i 114 . . 3  |-  ( A  e. FinIV  ->  -.  A  ~~  ( A  +c  1o ) )
35 brsdom 7089 . . 3  |-  ( A 
~<  ( A  +c  1o ) 
<->  ( A  ~<_  ( A  +c  1o )  /\  -.  A  ~~  ( A  +c  1o ) ) )
363, 34, 35sylanbrc 646 . 2  |-  ( A  e. FinIV  ->  A  ~<  ( A  +c  1o ) )
37 sdomnen 7095 . . . 4  |-  ( A 
~<  ( A  +c  1o )  ->  -.  A  ~~  ( A  +c  1o ) )
38 infcda1 8029 . . . . 5  |-  ( om  ~<_  A  ->  ( A  +c  1o )  ~~  A
)
3938ensymd 7117 . . . 4  |-  ( om  ~<_  A  ->  A  ~~  ( A  +c  1o ) )
4037, 39nsyl 115 . . 3  |-  ( A 
~<  ( A  +c  1o )  ->  -.  om  ~<_  A )
41 relsdom 7075 . . . . 5  |-  Rel  ~<
4241brrelexi 4877 . . . 4  |-  ( A 
~<  ( A  +c  1o )  ->  A  e.  _V )
43 isfin4-2 8150 . . . 4  |-  ( A  e.  _V  ->  ( A  e. FinIV 
<->  -.  om  ~<_  A ) )
4442, 43syl 16 . . 3  |-  ( A 
~<  ( A  +c  1o )  ->  ( A  e. FinIV  <->  -.  om  ~<_  A ) )
4540, 44mpbird 224 . 2  |-  ( A 
~<  ( A  +c  1o )  ->  A  e. FinIV )
4636, 45impbii 181 1  |-  ( A  e. FinIV  <-> 
A  ~<  ( A  +c  1o ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 177    = wceq 1649    e. wcel 1721    =/= wne 2567   _Vcvv 2916    u. cun 3278    C. wpss 3281   (/)c0 3588   {csn 3774   <.cop 3777   class class class wbr 4172   Oncon0 4541   omcom 4804    X. cxp 4835  (class class class)co 6040   1oc1o 6676    ~~ cen 7065    ~<_ cdom 7066    ~< csdm 7067    +c ccda 8003  FinIVcfin4 8116
This theorem is referenced by:  fin45  8228  finngch  8486  gchinf  8488
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-recs 6592  df-rdg 6627  df-1o 6683  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-cda 8004  df-fin4 8123
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