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Theorem fin1a2lem9 8779
Description: Lemma for fin1a2 8786. 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 7702 . . . . 5  |-  om  =  ( On  i^i  Fin )
2 inss2 3705 . . . . 5  |-  ( On 
i^i  Fin )  C_  Fin
31, 2eqsstri 3519 . . . 4  |-  om  C_  Fin
4 peano2 6693 . . . 4  |-  ( A  e.  om  ->  suc  A  e.  om )
53, 4sseldi 3487 . . 3  |-  ( A  e.  om  ->  suc  A  e.  Fin )
653ad2ant3 1017 . 2  |-  ( ( [ C.]  Or  X  /\  X  C_ 
Fin  /\  A  e.  om )  ->  suc  A  e. 
Fin )
743ad2ant3 1017 . . 3  |-  ( ( [ C.]  Or  X  /\  X  C_ 
Fin  /\  A  e.  om )  ->  suc  A  e. 
om )
8 breq1 4442 . . . . . 6  |-  ( b  =  c  ->  (
b  ~<_  A  <->  c  ~<_  A ) )
98elrab 3254 . . . . 5  |-  ( c  e.  { b  e.  X  |  b  ~<_  A }  <->  ( c  e.  X  /\  c  ~<_  A ) )
10 simprr 755 . . . . . . . 8  |-  ( ( ( [ C.]  Or  X  /\  X  C_  Fin  /\  A  e.  om )  /\  ( c  e.  X  /\  c  ~<_  A )
)  ->  c  ~<_  A )
11 simpl2 998 . . . . . . . . . . 11  |-  ( ( ( [ C.]  Or  X  /\  X  C_  Fin  /\  A  e.  om )  /\  ( c  e.  X  /\  c  ~<_  A )
)  ->  X  C_  Fin )
12 simprl 754 . . . . . . . . . . 11  |-  ( ( ( [ C.]  Or  X  /\  X  C_  Fin  /\  A  e.  om )  /\  ( c  e.  X  /\  c  ~<_  A )
)  ->  c  e.  X )
1311, 12sseldd 3490 . . . . . . . . . 10  |-  ( ( ( [ C.]  Or  X  /\  X  C_  Fin  /\  A  e.  om )  /\  ( c  e.  X  /\  c  ~<_  A )
)  ->  c  e.  Fin )
14 finnum 8320 . . . . . . . . . 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 999 . . . . . . . . . . 11  |-  ( ( ( [ C.]  Or  X  /\  X  C_  Fin  /\  A  e.  om )  /\  ( c  e.  X  /\  c  ~<_  A )
)  ->  A  e.  om )
173, 16sseldi 3487 . . . . . . . . . 10  |-  ( ( ( [ C.]  Or  X  /\  X  C_  Fin  /\  A  e.  om )  /\  ( c  e.  X  /\  c  ~<_  A )
)  ->  A  e.  Fin )
18 finnum 8320 . . . . . . . . . 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 8349 . . . . . . . . 9  |-  ( ( c  e.  dom  card  /\  A  e.  dom  card )  ->  ( ( card `  c )  C_  ( card `  A )  <->  c  ~<_  A ) )
2115, 19, 20syl2anc 659 . . . . . . . 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 432 . . . . . 6  |-  ( ( [ C.]  Or  X  /\  X  C_ 
Fin  /\  A  e.  om )  ->  ( (
c  e.  X  /\  c  ~<_  A )  -> 
( card `  c )  C_  ( card `  A
) ) )
24 cardnn 8335 . . . . . . . . 9  |-  ( A  e.  om  ->  ( card `  A )  =  A )
2524sseq2d 3517 . . . . . . . 8  |-  ( A  e.  om  ->  (
( card `  c )  C_  ( card `  A
)  <->  ( card `  c
)  C_  A )
)
26 cardon 8316 . . . . . . . . 9  |-  ( card `  c )  e.  On
27 nnon 6679 . . . . . . . . 9  |-  ( A  e.  om  ->  A  e.  On )
28 onsssuc 4954 . . . . . . . . 9  |-  ( ( ( card `  c
)  e.  On  /\  A  e.  On )  ->  ( ( card `  c
)  C_  A  <->  ( card `  c )  e.  suc  A ) )
2926, 27, 28sylancr 661 . . . . . . . 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 1017 . . . . . 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 3251 . . . . 5  |-  ( c  e.  { b  e.  X  |  b  ~<_  A }  ->  c  e.  X )
35 elrabi 3251 . . . . 5  |-  ( d  e.  { b  e.  X  |  b  ~<_  A }  ->  d  e.  X )
36 ssel 3483 . . . . . . . . . . 11  |-  ( X 
C_  Fin  ->  ( c  e.  X  ->  c  e.  Fin ) )
37 ssel 3483 . . . . . . . . . . 11  |-  ( X 
C_  Fin  ->  ( d  e.  X  ->  d  e.  Fin ) )
3836, 37anim12d 561 . . . . . . . . . 10  |-  ( X 
C_  Fin  ->  ( ( c  e.  X  /\  d  e.  X )  ->  ( c  e.  Fin  /\  d  e.  Fin )
) )
3938imp 427 . . . . . . . . 9  |-  ( ( X  C_  Fin  /\  (
c  e.  X  /\  d  e.  X )
)  ->  ( c  e.  Fin  /\  d  e. 
Fin ) )
40393ad2antl2 1157 . . . . . . . 8  |-  ( ( ( [ C.]  Or  X  /\  X  C_  Fin  /\  A  e.  om )  /\  ( c  e.  X  /\  d  e.  X
) )  ->  (
c  e.  Fin  /\  d  e.  Fin )
)
41 sorpssi 6559 . . . . . . . . 9  |-  ( ( [ C.]  Or  X  /\  (
c  e.  X  /\  d  e.  X )
)  ->  ( c  C_  d  \/  d  C_  c ) )
42413ad2antl1 1156 . . . . . . . 8  |-  ( ( ( [ C.]  Or  X  /\  X  C_  Fin  /\  A  e.  om )  /\  ( c  e.  X  /\  d  e.  X
) )  ->  (
c  C_  d  \/  d  C_  c ) )
43 finnum 8320 . . . . . . . . . . 11  |-  ( d  e.  Fin  ->  d  e.  dom  card )
44 carden2 8359 . . . . . . . . . . 11  |-  ( ( c  e.  dom  card  /\  d  e.  dom  card )  ->  ( ( card `  c )  =  (
card `  d )  <->  c 
~~  d ) )
4514, 43, 44syl2an 475 . . . . . . . . . 10  |-  ( ( c  e.  Fin  /\  d  e.  Fin )  ->  ( ( card `  c
)  =  ( card `  d )  <->  c  ~~  d ) )
4645adantr 463 . . . . . . . . 9  |-  ( ( ( c  e.  Fin  /\  d  e.  Fin )  /\  ( c  C_  d  \/  d  C_  c ) )  ->  ( ( card `  c )  =  ( card `  d
)  <->  c  ~~  d
) )
47 fin23lem25 8695 . . . . . . . . . . 11  |-  ( ( c  e.  Fin  /\  d  e.  Fin  /\  (
c  C_  d  \/  d  C_  c ) )  ->  ( c  ~~  d 
<->  c  =  d ) )
48473expa 1194 . . . . . . . . . 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 659 . . . . . . 7  |-  ( ( ( [ C.]  Or  X  /\  X  C_  Fin  /\  A  e.  om )  /\  ( c  e.  X  /\  d  e.  X
) )  ->  (
( card `  c )  =  ( card `  d
)  ->  c  =  d ) )
52 fveq2 5848 . . . . . . 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 432 . . . . 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 654 . . . 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 7549 . . 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 7734 . 2  |-  ( ( suc  A  e.  Fin  /\ 
{ b  e.  X  |  b  ~<_  A }  ~<_  suc  A )  ->  { b  e.  X  |  b  ~<_  A }  e.  Fin )
596, 57, 58syl2anc 659 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 366    /\ wa 367    /\ w3a 971    = wceq 1398    e. wcel 1823   {crab 2808    i^i cin 3460    C_ wss 3461   class class class wbr 4439    Or wor 4788   Oncon0 4867   suc csuc 4869   dom cdm 4988   ` cfv 5570   [ C.] crpss 6552   omcom 6673    ~~ cen 7506    ~<_ cdom 7507   Fincfn 7509   cardccrd 8307
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-rpss 6553  df-om 6674  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-card 8311
This theorem is referenced by:  fin1a2lem11  8781
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