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Theorem fin1a2lem9 8280
Description: Lemma for fin1a2 8287. In a chain of finite sets, initial segments are finite. (Contributed by Stefan O'Rear, 8-Nov-2014.)
Assertion
Ref Expression
fin1a2lem9  |-  ( ( [
C.]  Or  X  /\  X  C_  Fin  /\  A  e.  om )  ->  { b  e.  X  |  b  ~<_  A }  e.  Fin )
Distinct variable groups:    A, b    X, b

Proof of Theorem fin1a2lem9
Dummy variables  c 
d are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 onfin2 7290 . . . . 5  |-  om  =  ( On  i^i  Fin )
2 inss2 3554 . . . . 5  |-  ( On 
i^i  Fin )  C_  Fin
31, 2eqsstri 3370 . . . 4  |-  om  C_  Fin
4 peano2 4857 . . . 4  |-  ( A  e.  om  ->  suc  A  e.  om )
53, 4sseldi 3338 . . 3  |-  ( A  e.  om  ->  suc  A  e.  Fin )
653ad2ant3 980 . 2  |-  ( ( [
C.]  Or  X  /\  X  C_  Fin  /\  A  e.  om )  ->  suc  A  e.  Fin )
743ad2ant3 980 . . 3  |-  ( ( [
C.]  Or  X  /\  X  C_  Fin  /\  A  e.  om )  ->  suc  A  e.  om )
8 breq1 4207 . . . . . 6  |-  ( b  =  c  ->  (
b  ~<_  A  <->  c  ~<_  A ) )
98elrab 3084 . . . . 5  |-  ( c  e.  { b  e.  X  |  b  ~<_  A }  <->  ( c  e.  X  /\  c  ~<_  A ) )
10 simprr 734 . . . . . . . 8  |-  ( ( ( [ C.]  Or  X  /\  X  C_  Fin  /\  A  e.  om )  /\  ( c  e.  X  /\  c  ~<_  A )
)  ->  c  ~<_  A )
11 simpl2 961 . . . . . . . . . . 11  |-  ( ( ( [ C.]  Or  X  /\  X  C_  Fin  /\  A  e.  om )  /\  ( c  e.  X  /\  c  ~<_  A )
)  ->  X  C_  Fin )
12 simprl 733 . . . . . . . . . . 11  |-  ( ( ( [ C.]  Or  X  /\  X  C_  Fin  /\  A  e.  om )  /\  ( c  e.  X  /\  c  ~<_  A )
)  ->  c  e.  X )
1311, 12sseldd 3341 . . . . . . . . . 10  |-  ( ( ( [ C.]  Or  X  /\  X  C_  Fin  /\  A  e.  om )  /\  ( c  e.  X  /\  c  ~<_  A )
)  ->  c  e.  Fin )
14 finnum 7827 . . . . . . . . . 10  |-  ( c  e.  Fin  ->  c  e.  dom  card )
1513, 14syl 16 . . . . . . . . 9  |-  ( ( ( [ C.]  Or  X  /\  X  C_  Fin  /\  A  e.  om )  /\  ( c  e.  X  /\  c  ~<_  A )
)  ->  c  e.  dom  card )
16 simpl3 962 . . . . . . . . . . 11  |-  ( ( ( [ C.]  Or  X  /\  X  C_  Fin  /\  A  e.  om )  /\  ( c  e.  X  /\  c  ~<_  A )
)  ->  A  e.  om )
173, 16sseldi 3338 . . . . . . . . . 10  |-  ( ( ( [ C.]  Or  X  /\  X  C_  Fin  /\  A  e.  om )  /\  ( c  e.  X  /\  c  ~<_  A )
)  ->  A  e.  Fin )
18 finnum 7827 . . . . . . . . . 10  |-  ( A  e.  Fin  ->  A  e.  dom  card )
1917, 18syl 16 . . . . . . . . 9  |-  ( ( ( [ C.]  Or  X  /\  X  C_  Fin  /\  A  e.  om )  /\  ( c  e.  X  /\  c  ~<_  A )
)  ->  A  e.  dom  card )
20 carddom2 7856 . . . . . . . . 9  |-  ( ( c  e.  dom  card  /\  A  e.  dom  card )  ->  ( ( card `  c )  C_  ( card `  A )  <->  c  ~<_  A ) )
2115, 19, 20syl2anc 643 . . . . . . . 8  |-  ( ( ( [ C.]  Or  X  /\  X  C_  Fin  /\  A  e.  om )  /\  ( c  e.  X  /\  c  ~<_  A )
)  ->  ( ( card `  c )  C_  ( card `  A )  <->  c  ~<_  A ) )
2210, 21mpbird 224 . . . . . . 7  |-  ( ( ( [ C.]  Or  X  /\  X  C_  Fin  /\  A  e.  om )  /\  ( c  e.  X  /\  c  ~<_  A )
)  ->  ( card `  c )  C_  ( card `  A ) )
2322ex 424 . . . . . 6  |-  ( ( [
C.]  Or  X  /\  X  C_  Fin  /\  A  e.  om )  ->  (
( c  e.  X  /\  c  ~<_  A )  ->  ( card `  c
)  C_  ( card `  A ) ) )
24 cardnn 7842 . . . . . . . . 9  |-  ( A  e.  om  ->  ( card `  A )  =  A )
2524sseq2d 3368 . . . . . . . 8  |-  ( A  e.  om  ->  (
( card `  c )  C_  ( card `  A
)  <->  ( card `  c
)  C_  A )
)
26 cardon 7823 . . . . . . . . 9  |-  ( card `  c )  e.  On
27 nnon 4843 . . . . . . . . 9  |-  ( A  e.  om  ->  A  e.  On )
28 onsssuc 4661 . . . . . . . . 9  |-  ( ( ( card `  c
)  e.  On  /\  A  e.  On )  ->  ( ( card `  c
)  C_  A  <->  ( card `  c )  e.  suc  A ) )
2926, 27, 28sylancr 645 . . . . . . . 8  |-  ( A  e.  om  ->  (
( card `  c )  C_  A  <->  ( card `  c
)  e.  suc  A
) )
3025, 29bitrd 245 . . . . . . 7  |-  ( A  e.  om  ->  (
( card `  c )  C_  ( card `  A
)  <->  ( card `  c
)  e.  suc  A
) )
31303ad2ant3 980 . . . . . 6  |-  ( ( [
C.]  Or  X  /\  X  C_  Fin  /\  A  e.  om )  ->  (
( card `  c )  C_  ( card `  A
)  <->  ( card `  c
)  e.  suc  A
) )
3223, 31sylibd 206 . . . . 5  |-  ( ( [
C.]  Or  X  /\  X  C_  Fin  /\  A  e.  om )  ->  (
( c  e.  X  /\  c  ~<_  A )  ->  ( card `  c
)  e.  suc  A
) )
339, 32syl5bi 209 . . . 4  |-  ( ( [
C.]  Or  X  /\  X  C_  Fin  /\  A  e.  om )  ->  (
c  e.  { b  e.  X  |  b  ~<_  A }  ->  ( card `  c )  e. 
suc  A ) )
34 elrabi 3082 . . . . 5  |-  ( c  e.  { b  e.  X  |  b  ~<_  A }  ->  c  e.  X )
35 elrabi 3082 . . . . 5  |-  ( d  e.  { b  e.  X  |  b  ~<_  A }  ->  d  e.  X )
36 ssel 3334 . . . . . . . . . . 11  |-  ( X 
C_  Fin  ->  ( c  e.  X  ->  c  e.  Fin ) )
37 ssel 3334 . . . . . . . . . . 11  |-  ( X 
C_  Fin  ->  ( d  e.  X  ->  d  e.  Fin ) )
3836, 37anim12d 547 . . . . . . . . . 10  |-  ( X 
C_  Fin  ->  ( ( c  e.  X  /\  d  e.  X )  ->  ( c  e.  Fin  /\  d  e.  Fin )
) )
3938imp 419 . . . . . . . . 9  |-  ( ( X  C_  Fin  /\  (
c  e.  X  /\  d  e.  X )
)  ->  ( c  e.  Fin  /\  d  e. 
Fin ) )
40393ad2antl2 1120 . . . . . . . 8  |-  ( ( ( [ C.]  Or  X  /\  X  C_  Fin  /\  A  e.  om )  /\  ( c  e.  X  /\  d  e.  X
) )  ->  (
c  e.  Fin  /\  d  e.  Fin )
)
41 sorpssi 6520 . . . . . . . . 9  |-  ( ( [
C.]  Or  X  /\  ( c  e.  X  /\  d  e.  X
) )  ->  (
c  C_  d  \/  d  C_  c ) )
42413ad2antl1 1119 . . . . . . . 8  |-  ( ( ( [ C.]  Or  X  /\  X  C_  Fin  /\  A  e.  om )  /\  ( c  e.  X  /\  d  e.  X
) )  ->  (
c  C_  d  \/  d  C_  c ) )
43 finnum 7827 . . . . . . . . . . 11  |-  ( d  e.  Fin  ->  d  e.  dom  card )
44 carden2 7866 . . . . . . . . . . 11  |-  ( ( c  e.  dom  card  /\  d  e.  dom  card )  ->  ( ( card `  c )  =  (
card `  d )  <->  c 
~~  d ) )
4514, 43, 44syl2an 464 . . . . . . . . . 10  |-  ( ( c  e.  Fin  /\  d  e.  Fin )  ->  ( ( card `  c
)  =  ( card `  d )  <->  c  ~~  d ) )
4645adantr 452 . . . . . . . . 9  |-  ( ( ( c  e.  Fin  /\  d  e.  Fin )  /\  ( c  C_  d  \/  d  C_  c ) )  ->  ( ( card `  c )  =  ( card `  d
)  <->  c  ~~  d
) )
47 fin23lem25 8196 . . . . . . . . . . 11  |-  ( ( c  e.  Fin  /\  d  e.  Fin  /\  (
c  C_  d  \/  d  C_  c ) )  ->  ( c  ~~  d 
<->  c  =  d ) )
48473expa 1153 . . . . . . . . . 10  |-  ( ( ( c  e.  Fin  /\  d  e.  Fin )  /\  ( c  C_  d  \/  d  C_  c ) )  ->  ( c  ~~  d  <->  c  =  d ) )
4948biimpd 199 . . . . . . . . 9  |-  ( ( ( c  e.  Fin  /\  d  e.  Fin )  /\  ( c  C_  d  \/  d  C_  c ) )  ->  ( c  ~~  d  ->  c  =  d ) )
5046, 49sylbid 207 . . . . . . . 8  |-  ( ( ( c  e.  Fin  /\  d  e.  Fin )  /\  ( c  C_  d  \/  d  C_  c ) )  ->  ( ( card `  c )  =  ( card `  d
)  ->  c  =  d ) )
5140, 42, 50syl2anc 643 . . . . . . 7  |-  ( ( ( [ C.]  Or  X  /\  X  C_  Fin  /\  A  e.  om )  /\  ( c  e.  X  /\  d  e.  X
) )  ->  (
( card `  c )  =  ( card `  d
)  ->  c  =  d ) )
52 fveq2 5720 . . . . . . 7  |-  ( c  =  d  ->  ( card `  c )  =  ( card `  d
) )
5351, 52impbid1 195 . . . . . 6  |-  ( ( ( [ C.]  Or  X  /\  X  C_  Fin  /\  A  e.  om )  /\  ( c  e.  X  /\  d  e.  X
) )  ->  (
( card `  c )  =  ( card `  d
)  <->  c  =  d ) )
5453ex 424 . . . . 5  |-  ( ( [
C.]  Or  X  /\  X  C_  Fin  /\  A  e.  om )  ->  (
( c  e.  X  /\  d  e.  X
)  ->  ( ( card `  c )  =  ( card `  d
)  <->  c  =  d ) ) )
5534, 35, 54syl2ani 638 . . . 4  |-  ( ( [
C.]  Or  X  /\  X  C_  Fin  /\  A  e.  om )  ->  (
( c  e.  {
b  e.  X  | 
b  ~<_  A }  /\  d  e.  { b  e.  X  |  b  ~<_  A } )  ->  (
( card `  c )  =  ( card `  d
)  <->  c  =  d ) ) )
5633, 55dom2d 7140 . . 3  |-  ( ( [
C.]  Or  X  /\  X  C_  Fin  /\  A  e.  om )  ->  ( suc  A  e.  om  ->  { b  e.  X  | 
b  ~<_  A }  ~<_  suc  A
) )
577, 56mpd 15 . 2  |-  ( ( [
C.]  Or  X  /\  X  C_  Fin  /\  A  e.  om )  ->  { b  e.  X  |  b  ~<_  A }  ~<_  suc  A
)
58 domfi 7322 . 2  |-  ( ( suc  A  e.  Fin  /\ 
{ b  e.  X  |  b  ~<_  A }  ~<_  suc  A )  ->  { b  e.  X  |  b  ~<_  A }  e.  Fin )
596, 57, 58syl2anc 643 1  |-  ( ( [
C.]  Or  X  /\  X  C_  Fin  /\  A  e.  om )  ->  { b  e.  X  |  b  ~<_  A }  e.  Fin )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   {crab 2701    i^i cin 3311    C_ wss 3312   class class class wbr 4204    Or wor 4494   Oncon0 4573   suc csuc 4575   omcom 4837   dom cdm 4870   ` cfv 5446   [ C.] crpss 6513    ~~ cen 7098    ~<_ cdom 7099   Fincfn 7101   cardccrd 7814
This theorem is referenced by:  fin1a2lem11  8282
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-rpss 6514  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-card 7818
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