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Theorem fin1a2lem9 8683
Description: Lemma for fin1a2 8690. 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 7608 . . . . 5  |-  om  =  ( On  i^i  Fin )
2 inss2 3674 . . . . 5  |-  ( On 
i^i  Fin )  C_  Fin
31, 2eqsstri 3489 . . . 4  |-  om  C_  Fin
4 peano2 6601 . . . 4  |-  ( A  e.  om  ->  suc  A  e.  om )
53, 4sseldi 3457 . . 3  |-  ( A  e.  om  ->  suc  A  e.  Fin )
653ad2ant3 1011 . 2  |-  ( ( [
C.]  Or  X  /\  X  C_  Fin  /\  A  e.  om )  ->  suc  A  e.  Fin )
743ad2ant3 1011 . . 3  |-  ( ( [
C.]  Or  X  /\  X  C_  Fin  /\  A  e.  om )  ->  suc  A  e.  om )
8 breq1 4398 . . . . . 6  |-  ( b  =  c  ->  (
b  ~<_  A  <->  c  ~<_  A ) )
98elrab 3218 . . . . 5  |-  ( c  e.  { b  e.  X  |  b  ~<_  A }  <->  ( c  e.  X  /\  c  ~<_  A ) )
10 simprr 756 . . . . . . . 8  |-  ( ( ( [ C.]  Or  X  /\  X  C_  Fin  /\  A  e.  om )  /\  ( c  e.  X  /\  c  ~<_  A )
)  ->  c  ~<_  A )
11 simpl2 992 . . . . . . . . . . 11  |-  ( ( ( [ C.]  Or  X  /\  X  C_  Fin  /\  A  e.  om )  /\  ( c  e.  X  /\  c  ~<_  A )
)  ->  X  C_  Fin )
12 simprl 755 . . . . . . . . . . 11  |-  ( ( ( [ C.]  Or  X  /\  X  C_  Fin  /\  A  e.  om )  /\  ( c  e.  X  /\  c  ~<_  A )
)  ->  c  e.  X )
1311, 12sseldd 3460 . . . . . . . . . 10  |-  ( ( ( [ C.]  Or  X  /\  X  C_  Fin  /\  A  e.  om )  /\  ( c  e.  X  /\  c  ~<_  A )
)  ->  c  e.  Fin )
14 finnum 8224 . . . . . . . . . 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 993 . . . . . . . . . . 11  |-  ( ( ( [ C.]  Or  X  /\  X  C_  Fin  /\  A  e.  om )  /\  ( c  e.  X  /\  c  ~<_  A )
)  ->  A  e.  om )
173, 16sseldi 3457 . . . . . . . . . 10  |-  ( ( ( [ C.]  Or  X  /\  X  C_  Fin  /\  A  e.  om )  /\  ( c  e.  X  /\  c  ~<_  A )
)  ->  A  e.  Fin )
18 finnum 8224 . . . . . . . . . 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 8253 . . . . . . . . 9  |-  ( ( c  e.  dom  card  /\  A  e.  dom  card )  ->  ( ( card `  c )  C_  ( card `  A )  <->  c  ~<_  A ) )
2115, 19, 20syl2anc 661 . . . . . . . 8  |-  ( ( ( [ C.]  Or  X  /\  X  C_  Fin  /\  A  e.  om )  /\  ( c  e.  X  /\  c  ~<_  A )
)  ->  ( ( card `  c )  C_  ( card `  A )  <->  c  ~<_  A ) )
2210, 21mpbird 232 . . . . . . 7  |-  ( ( ( [ C.]  Or  X  /\  X  C_  Fin  /\  A  e.  om )  /\  ( c  e.  X  /\  c  ~<_  A )
)  ->  ( card `  c )  C_  ( card `  A ) )
2322ex 434 . . . . . 6  |-  ( ( [
C.]  Or  X  /\  X  C_  Fin  /\  A  e.  om )  ->  (
( c  e.  X  /\  c  ~<_  A )  ->  ( card `  c
)  C_  ( card `  A ) ) )
24 cardnn 8239 . . . . . . . . 9  |-  ( A  e.  om  ->  ( card `  A )  =  A )
2524sseq2d 3487 . . . . . . . 8  |-  ( A  e.  om  ->  (
( card `  c )  C_  ( card `  A
)  <->  ( card `  c
)  C_  A )
)
26 cardon 8220 . . . . . . . . 9  |-  ( card `  c )  e.  On
27 nnon 6587 . . . . . . . . 9  |-  ( A  e.  om  ->  A  e.  On )
28 onsssuc 4909 . . . . . . . . 9  |-  ( ( ( card `  c
)  e.  On  /\  A  e.  On )  ->  ( ( card `  c
)  C_  A  <->  ( card `  c )  e.  suc  A ) )
2926, 27, 28sylancr 663 . . . . . . . 8  |-  ( A  e.  om  ->  (
( card `  c )  C_  A  <->  ( card `  c
)  e.  suc  A
) )
3025, 29bitrd 253 . . . . . . 7  |-  ( A  e.  om  ->  (
( card `  c )  C_  ( card `  A
)  <->  ( card `  c
)  e.  suc  A
) )
31303ad2ant3 1011 . . . . . 6  |-  ( ( [
C.]  Or  X  /\  X  C_  Fin  /\  A  e.  om )  ->  (
( card `  c )  C_  ( card `  A
)  <->  ( card `  c
)  e.  suc  A
) )
3223, 31sylibd 214 . . . . 5  |-  ( ( [
C.]  Or  X  /\  X  C_  Fin  /\  A  e.  om )  ->  (
( c  e.  X  /\  c  ~<_  A )  ->  ( card `  c
)  e.  suc  A
) )
339, 32syl5bi 217 . . . 4  |-  ( ( [
C.]  Or  X  /\  X  C_  Fin  /\  A  e.  om )  ->  (
c  e.  { b  e.  X  |  b  ~<_  A }  ->  ( card `  c )  e. 
suc  A ) )
34 elrabi 3215 . . . . 5  |-  ( c  e.  { b  e.  X  |  b  ~<_  A }  ->  c  e.  X )
35 elrabi 3215 . . . . 5  |-  ( d  e.  { b  e.  X  |  b  ~<_  A }  ->  d  e.  X )
36 ssel 3453 . . . . . . . . . . 11  |-  ( X 
C_  Fin  ->  ( c  e.  X  ->  c  e.  Fin ) )
37 ssel 3453 . . . . . . . . . . 11  |-  ( X 
C_  Fin  ->  ( d  e.  X  ->  d  e.  Fin ) )
3836, 37anim12d 563 . . . . . . . . . 10  |-  ( X 
C_  Fin  ->  ( ( c  e.  X  /\  d  e.  X )  ->  ( c  e.  Fin  /\  d  e.  Fin )
) )
3938imp 429 . . . . . . . . 9  |-  ( ( X  C_  Fin  /\  (
c  e.  X  /\  d  e.  X )
)  ->  ( c  e.  Fin  /\  d  e. 
Fin ) )
40393ad2antl2 1151 . . . . . . . 8  |-  ( ( ( [ C.]  Or  X  /\  X  C_  Fin  /\  A  e.  om )  /\  ( c  e.  X  /\  d  e.  X
) )  ->  (
c  e.  Fin  /\  d  e.  Fin )
)
41 sorpssi 6471 . . . . . . . . 9  |-  ( ( [
C.]  Or  X  /\  ( c  e.  X  /\  d  e.  X
) )  ->  (
c  C_  d  \/  d  C_  c ) )
42413ad2antl1 1150 . . . . . . . 8  |-  ( ( ( [ C.]  Or  X  /\  X  C_  Fin  /\  A  e.  om )  /\  ( c  e.  X  /\  d  e.  X
) )  ->  (
c  C_  d  \/  d  C_  c ) )
43 finnum 8224 . . . . . . . . . . 11  |-  ( d  e.  Fin  ->  d  e.  dom  card )
44 carden2 8263 . . . . . . . . . . 11  |-  ( ( c  e.  dom  card  /\  d  e.  dom  card )  ->  ( ( card `  c )  =  (
card `  d )  <->  c 
~~  d ) )
4514, 43, 44syl2an 477 . . . . . . . . . 10  |-  ( ( c  e.  Fin  /\  d  e.  Fin )  ->  ( ( card `  c
)  =  ( card `  d )  <->  c  ~~  d ) )
4645adantr 465 . . . . . . . . 9  |-  ( ( ( c  e.  Fin  /\  d  e.  Fin )  /\  ( c  C_  d  \/  d  C_  c ) )  ->  ( ( card `  c )  =  ( card `  d
)  <->  c  ~~  d
) )
47 fin23lem25 8599 . . . . . . . . . . 11  |-  ( ( c  e.  Fin  /\  d  e.  Fin  /\  (
c  C_  d  \/  d  C_  c ) )  ->  ( c  ~~  d 
<->  c  =  d ) )
48473expa 1188 . . . . . . . . . 10  |-  ( ( ( c  e.  Fin  /\  d  e.  Fin )  /\  ( c  C_  d  \/  d  C_  c ) )  ->  ( c  ~~  d  <->  c  =  d ) )
4948biimpd 207 . . . . . . . . 9  |-  ( ( ( c  e.  Fin  /\  d  e.  Fin )  /\  ( c  C_  d  \/  d  C_  c ) )  ->  ( c  ~~  d  ->  c  =  d ) )
5046, 49sylbid 215 . . . . . . . 8  |-  ( ( ( c  e.  Fin  /\  d  e.  Fin )  /\  ( c  C_  d  \/  d  C_  c ) )  ->  ( ( card `  c )  =  ( card `  d
)  ->  c  =  d ) )
5140, 42, 50syl2anc 661 . . . . . . 7  |-  ( ( ( [ C.]  Or  X  /\  X  C_  Fin  /\  A  e.  om )  /\  ( c  e.  X  /\  d  e.  X
) )  ->  (
( card `  c )  =  ( card `  d
)  ->  c  =  d ) )
52 fveq2 5794 . . . . . . 7  |-  ( c  =  d  ->  ( card `  c )  =  ( card `  d
) )
5351, 52impbid1 203 . . . . . 6  |-  ( ( ( [ C.]  Or  X  /\  X  C_  Fin  /\  A  e.  om )  /\  ( c  e.  X  /\  d  e.  X
) )  ->  (
( card `  c )  =  ( card `  d
)  <->  c  =  d ) )
5453ex 434 . . . . 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 656 . . . 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 7455 . . 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 7640 . 2  |-  ( ( suc  A  e.  Fin  /\ 
{ b  e.  X  |  b  ~<_  A }  ~<_  suc  A )  ->  { b  e.  X  |  b  ~<_  A }  e.  Fin )
596, 57, 58syl2anc 661 1  |-  ( ( [
C.]  Or  X  /\  X  C_  Fin  /\  A  e.  om )  ->  { b  e.  X  |  b  ~<_  A }  e.  Fin )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758   {crab 2800    i^i cin 3430    C_ wss 3431   class class class wbr 4395    Or wor 4743   Oncon0 4822   suc csuc 4824   dom cdm 4943   ` cfv 5521   [ C.] crpss 6464   omcom 6581    ~~ cen 7412    ~<_ cdom 7413   Fincfn 7415   cardccrd 8211
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-ral 2801  df-rex 2802  df-reu 2803  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-pss 3447  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4195  df-int 4232  df-iun 4276  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4489  df-eprel 4735  df-id 4739  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-ord 4825  df-on 4826  df-lim 4827  df-suc 4828  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-rpss 6465  df-om 6582  df-er 7206  df-en 7416  df-dom 7417  df-sdom 7418  df-fin 7419  df-card 8215
This theorem is referenced by:  fin1a2lem11  8685
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