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Theorem isfin4-3 8159
Description: Alternate definition of IV-finite sets: they are strictly dominated by their successors. (Thus, the proper subset referred to in isfin4 8141 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 6698 . . . 4  |-  1o  e.  On
2 cdadom3 8032 . . . 4  |-  ( ( A  e. FinIV  /\  1o  e.  On )  ->  A  ~<_  ( A  +c  1o ) )
31, 2mpan2 653 . . 3  |-  ( A  e. FinIV  ->  A  ~<_  ( A  +c  1o ) )
4 ssun1 3478 . . . . . . . 8  |-  ( A  X.  { (/) } ) 
C_  ( ( A  X.  { (/) } )  u.  ( 1o  X.  { 1o } ) )
5 relen 7081 . . . . . . . . . 10  |-  Rel  ~~
65brrelexi 4885 . . . . . . . . 9  |-  ( A 
~~  ( A  +c  1o )  ->  A  e. 
_V )
7 cdaval 8014 . . . . . . . . 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 3365 . . . . . . 7  |-  ( A 
~~  ( A  +c  1o )  ->  ( A  X.  { (/) } ) 
C_  ( A  +c  1o ) )
10 0lt1o 6715 . . . . . . . . . 10  |-  (/)  e.  1o
111elexi 2933 . . . . . . . . . . 11  |-  1o  e.  _V
1211snid 3809 . . . . . . . . . 10  |-  1o  e.  { 1o }
13 opelxpi 4877 . . . . . . . . . 10  |-  ( (
(/)  e.  1o  /\  1o  e.  { 1o } )  ->  <. (/) ,  1o >.  e.  ( 1o  X.  { 1o } ) )
1410, 12, 13mp2an 654 . . . . . . . . 9  |-  <. (/) ,  1o >.  e.  ( 1o  X.  { 1o } )
15 elun2 3483 . . . . . . . . 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 2489 . . . . . . 7  |-  ( A 
~~  ( A  +c  1o )  ->  <. (/) ,  1o >.  e.  ( A  +c  1o ) )
18 1n0 6706 . . . . . . . 8  |-  1o  =/=  (/)
19 opelxp2 4879 . . . . . . . . . 10  |-  ( <. (/)
,  1o >.  e.  ( A  X.  { (/) } )  ->  1o  e.  {
(/) } )
20 elsni 3806 . . . . . . . . . 10  |-  ( 1o  e.  { (/) }  ->  1o  =  (/) )
2119, 20syl 16 . . . . . . . . 9  |-  ( <. (/)
,  1o >.  e.  ( A  X.  { (/) } )  ->  1o  =  (/) )
2221necon3ai 2615 . . . . . . . 8  |-  ( 1o  =/=  (/)  ->  -.  <. (/) ,  1o >.  e.  ( A  X.  { (/) } ) )
2318, 22mp1i 12 . . . . . . 7  |-  ( A 
~~  ( A  +c  1o )  ->  -.  <. (/)
,  1o >.  e.  ( A  X.  { (/) } ) )
249, 17, 23ssnelpssd 3660 . . . . . 6  |-  ( A 
~~  ( A  +c  1o )  ->  ( A  X.  { (/) } ) 
C.  ( A  +c  1o ) )
25 0ex 4307 . . . . . . . 8  |-  (/)  e.  _V
26 xpsneng 7160 . . . . . . . 8  |-  ( ( A  e.  _V  /\  (/) 
e.  _V )  ->  ( A  X.  { (/) } ) 
~~  A )
276, 25, 26sylancl 644 . . . . . . 7  |-  ( A 
~~  ( A  +c  1o )  ->  ( A  X.  { (/) } ) 
~~  A )
28 entr 7126 . . . . . . 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 8142 . . . . . 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 8153 . . . . 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 7097 . . 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 7103 . . . 4  |-  ( A 
~<  ( A  +c  1o )  ->  -.  A  ~~  ( A  +c  1o ) )
38 infcda1 8037 . . . . 5  |-  ( om  ~<_  A  ->  ( A  +c  1o )  ~~  A
)
3938ensymd 7125 . . . 4  |-  ( om  ~<_  A  ->  A  ~~  ( A  +c  1o ) )
4037, 39nsyl 115 . . 3  |-  ( A 
~<  ( A  +c  1o )  ->  -.  om  ~<_  A )
41 relsdom 7083 . . . . 5  |-  Rel  ~<
4241brrelexi 4885 . . . 4  |-  ( A 
~<  ( A  +c  1o )  ->  A  e.  _V )
43 isfin4-2 8158 . . . 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 2575   _Vcvv 2924    u. cun 3286    C. wpss 3289   (/)c0 3596   {csn 3782   <.cop 3785   class class class wbr 4180   Oncon0 4549   omcom 4812    X. cxp 4843  (class class class)co 6048   1oc1o 6684    ~~ cen 7073    ~<_ cdom 7074    ~< csdm 7075    +c ccda 8011  FinIVcfin4 8124
This theorem is referenced by:  fin45  8236  finngch  8494  gchinf  8496
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 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668
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 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-reu 2681  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-int 4019  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  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 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-recs 6600  df-rdg 6635  df-1o 6691  df-er 6872  df-en 7077  df-dom 7078  df-sdom 7079  df-fin 7080  df-cda 8012  df-fin4 8131
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