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Theorem fin1a2lem9 8050
Description: Lemma for fin1a2 8057. 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 7068 . . . . 5  |-  om  =  ( On  i^i  Fin )
2 inss2 3403 . . . . 5  |-  ( On 
i^i  Fin )  C_  Fin
31, 2eqsstri 3221 . . . 4  |-  om  C_  Fin
4 peano2 4692 . . . 4  |-  ( A  e.  om  ->  suc  A  e.  om )
53, 4sseldi 3191 . . 3  |-  ( A  e.  om  ->  suc  A  e.  Fin )
653ad2ant3 978 . 2  |-  ( ( [
C.]  Or  X  /\  X  C_  Fin  /\  A  e.  om )  ->  suc  A  e.  Fin )
743ad2ant3 978 . . 3  |-  ( ( [
C.]  Or  X  /\  X  C_  Fin  /\  A  e.  om )  ->  suc  A  e.  om )
8 breq1 4042 . . . . . 6  |-  ( b  =  c  ->  (
b  ~<_  A  <->  c  ~<_  A ) )
98elrab 2936 . . . . 5  |-  ( c  e.  { b  e.  X  |  b  ~<_  A }  <->  ( c  e.  X  /\  c  ~<_  A ) )
10 simprr 733 . . . . . . . 8  |-  ( ( ( [ C.]  Or  X  /\  X  C_  Fin  /\  A  e.  om )  /\  ( c  e.  X  /\  c  ~<_  A )
)  ->  c  ~<_  A )
11 simpl2 959 . . . . . . . . . . 11  |-  ( ( ( [ C.]  Or  X  /\  X  C_  Fin  /\  A  e.  om )  /\  ( c  e.  X  /\  c  ~<_  A )
)  ->  X  C_  Fin )
12 simprl 732 . . . . . . . . . . 11  |-  ( ( ( [ C.]  Or  X  /\  X  C_  Fin  /\  A  e.  om )  /\  ( c  e.  X  /\  c  ~<_  A )
)  ->  c  e.  X )
1311, 12sseldd 3194 . . . . . . . . . 10  |-  ( ( ( [ C.]  Or  X  /\  X  C_  Fin  /\  A  e.  om )  /\  ( c  e.  X  /\  c  ~<_  A )
)  ->  c  e.  Fin )
14 finnum 7597 . . . . . . . . . 10  |-  ( c  e.  Fin  ->  c  e.  dom  card )
1513, 14syl 15 . . . . . . . . 9  |-  ( ( ( [ C.]  Or  X  /\  X  C_  Fin  /\  A  e.  om )  /\  ( c  e.  X  /\  c  ~<_  A )
)  ->  c  e.  dom  card )
16 simpl3 960 . . . . . . . . . . 11  |-  ( ( ( [ C.]  Or  X  /\  X  C_  Fin  /\  A  e.  om )  /\  ( c  e.  X  /\  c  ~<_  A )
)  ->  A  e.  om )
173, 16sseldi 3191 . . . . . . . . . 10  |-  ( ( ( [ C.]  Or  X  /\  X  C_  Fin  /\  A  e.  om )  /\  ( c  e.  X  /\  c  ~<_  A )
)  ->  A  e.  Fin )
18 finnum 7597 . . . . . . . . . 10  |-  ( A  e.  Fin  ->  A  e.  dom  card )
1917, 18syl 15 . . . . . . . . 9  |-  ( ( ( [ C.]  Or  X  /\  X  C_  Fin  /\  A  e.  om )  /\  ( c  e.  X  /\  c  ~<_  A )
)  ->  A  e.  dom  card )
20 carddom2 7626 . . . . . . . . 9  |-  ( ( c  e.  dom  card  /\  A  e.  dom  card )  ->  ( ( card `  c )  C_  ( card `  A )  <->  c  ~<_  A ) )
2115, 19, 20syl2anc 642 . . . . . . . 8  |-  ( ( ( [ C.]  Or  X  /\  X  C_  Fin  /\  A  e.  om )  /\  ( c  e.  X  /\  c  ~<_  A )
)  ->  ( ( card `  c )  C_  ( card `  A )  <->  c  ~<_  A ) )
2210, 21mpbird 223 . . . . . . 7  |-  ( ( ( [ C.]  Or  X  /\  X  C_  Fin  /\  A  e.  om )  /\  ( c  e.  X  /\  c  ~<_  A )
)  ->  ( card `  c )  C_  ( card `  A ) )
2322ex 423 . . . . . 6  |-  ( ( [
C.]  Or  X  /\  X  C_  Fin  /\  A  e.  om )  ->  (
( c  e.  X  /\  c  ~<_  A )  ->  ( card `  c
)  C_  ( card `  A ) ) )
24 cardnn 7612 . . . . . . . . 9  |-  ( A  e.  om  ->  ( card `  A )  =  A )
2524sseq2d 3219 . . . . . . . 8  |-  ( A  e.  om  ->  (
( card `  c )  C_  ( card `  A
)  <->  ( card `  c
)  C_  A )
)
26 cardon 7593 . . . . . . . . 9  |-  ( card `  c )  e.  On
27 nnon 4678 . . . . . . . . 9  |-  ( A  e.  om  ->  A  e.  On )
28 onsssuc 4496 . . . . . . . . 9  |-  ( ( ( card `  c
)  e.  On  /\  A  e.  On )  ->  ( ( card `  c
)  C_  A  <->  ( card `  c )  e.  suc  A ) )
2926, 27, 28sylancr 644 . . . . . . . 8  |-  ( A  e.  om  ->  (
( card `  c )  C_  A  <->  ( card `  c
)  e.  suc  A
) )
3025, 29bitrd 244 . . . . . . 7  |-  ( A  e.  om  ->  (
( card `  c )  C_  ( card `  A
)  <->  ( card `  c
)  e.  suc  A
) )
31303ad2ant3 978 . . . . . 6  |-  ( ( [
C.]  Or  X  /\  X  C_  Fin  /\  A  e.  om )  ->  (
( card `  c )  C_  ( card `  A
)  <->  ( card `  c
)  e.  suc  A
) )
3223, 31sylibd 205 . . . . 5  |-  ( ( [
C.]  Or  X  /\  X  C_  Fin  /\  A  e.  om )  ->  (
( c  e.  X  /\  c  ~<_  A )  ->  ( card `  c
)  e.  suc  A
) )
339, 32syl5bi 208 . . . 4  |-  ( ( [
C.]  Or  X  /\  X  C_  Fin  /\  A  e.  om )  ->  (
c  e.  { b  e.  X  |  b  ~<_  A }  ->  ( card `  c )  e. 
suc  A ) )
34 ssrab2 3271 . . . . . 6  |-  { b  e.  X  |  b  ~<_  A }  C_  X
3534sseli 3189 . . . . 5  |-  ( c  e.  { b  e.  X  |  b  ~<_  A }  ->  c  e.  X )
3634sseli 3189 . . . . 5  |-  ( d  e.  { b  e.  X  |  b  ~<_  A }  ->  d  e.  X )
37 ssel 3187 . . . . . . . . . . 11  |-  ( X 
C_  Fin  ->  ( c  e.  X  ->  c  e.  Fin ) )
38 ssel 3187 . . . . . . . . . . 11  |-  ( X 
C_  Fin  ->  ( d  e.  X  ->  d  e.  Fin ) )
3937, 38anim12d 546 . . . . . . . . . 10  |-  ( X 
C_  Fin  ->  ( ( c  e.  X  /\  d  e.  X )  ->  ( c  e.  Fin  /\  d  e.  Fin )
) )
4039imp 418 . . . . . . . . 9  |-  ( ( X  C_  Fin  /\  (
c  e.  X  /\  d  e.  X )
)  ->  ( c  e.  Fin  /\  d  e. 
Fin ) )
41403ad2antl2 1118 . . . . . . . 8  |-  ( ( ( [ C.]  Or  X  /\  X  C_  Fin  /\  A  e.  om )  /\  ( c  e.  X  /\  d  e.  X
) )  ->  (
c  e.  Fin  /\  d  e.  Fin )
)
42 sorpssi 6299 . . . . . . . . 9  |-  ( ( [
C.]  Or  X  /\  ( c  e.  X  /\  d  e.  X
) )  ->  (
c  C_  d  \/  d  C_  c ) )
43423ad2antl1 1117 . . . . . . . 8  |-  ( ( ( [ C.]  Or  X  /\  X  C_  Fin  /\  A  e.  om )  /\  ( c  e.  X  /\  d  e.  X
) )  ->  (
c  C_  d  \/  d  C_  c ) )
44 finnum 7597 . . . . . . . . . . 11  |-  ( d  e.  Fin  ->  d  e.  dom  card )
45 carden2 7636 . . . . . . . . . . 11  |-  ( ( c  e.  dom  card  /\  d  e.  dom  card )  ->  ( ( card `  c )  =  (
card `  d )  <->  c 
~~  d ) )
4614, 44, 45syl2an 463 . . . . . . . . . 10  |-  ( ( c  e.  Fin  /\  d  e.  Fin )  ->  ( ( card `  c
)  =  ( card `  d )  <->  c  ~~  d ) )
4746adantr 451 . . . . . . . . 9  |-  ( ( ( c  e.  Fin  /\  d  e.  Fin )  /\  ( c  C_  d  \/  d  C_  c ) )  ->  ( ( card `  c )  =  ( card `  d
)  <->  c  ~~  d
) )
48 fin23lem25 7966 . . . . . . . . . . 11  |-  ( ( c  e.  Fin  /\  d  e.  Fin  /\  (
c  C_  d  \/  d  C_  c ) )  ->  ( c  ~~  d 
<->  c  =  d ) )
49483expa 1151 . . . . . . . . . 10  |-  ( ( ( c  e.  Fin  /\  d  e.  Fin )  /\  ( c  C_  d  \/  d  C_  c ) )  ->  ( c  ~~  d  <->  c  =  d ) )
5049biimpd 198 . . . . . . . . 9  |-  ( ( ( c  e.  Fin  /\  d  e.  Fin )  /\  ( c  C_  d  \/  d  C_  c ) )  ->  ( c  ~~  d  ->  c  =  d ) )
5147, 50sylbid 206 . . . . . . . 8  |-  ( ( ( c  e.  Fin  /\  d  e.  Fin )  /\  ( c  C_  d  \/  d  C_  c ) )  ->  ( ( card `  c )  =  ( card `  d
)  ->  c  =  d ) )
5241, 43, 51syl2anc 642 . . . . . . 7  |-  ( ( ( [ C.]  Or  X  /\  X  C_  Fin  /\  A  e.  om )  /\  ( c  e.  X  /\  d  e.  X
) )  ->  (
( card `  c )  =  ( card `  d
)  ->  c  =  d ) )
53 fveq2 5541 . . . . . . 7  |-  ( c  =  d  ->  ( card `  c )  =  ( card `  d
) )
5452, 53impbid1 194 . . . . . 6  |-  ( ( ( [ C.]  Or  X  /\  X  C_  Fin  /\  A  e.  om )  /\  ( c  e.  X  /\  d  e.  X
) )  ->  (
( card `  c )  =  ( card `  d
)  <->  c  =  d ) )
5554ex 423 . . . . 5  |-  ( ( [
C.]  Or  X  /\  X  C_  Fin  /\  A  e.  om )  ->  (
( c  e.  X  /\  d  e.  X
)  ->  ( ( card `  c )  =  ( card `  d
)  <->  c  =  d ) ) )
5635, 36, 55syl2ani 637 . . . 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 ) ) )
5733, 56dom2d 6918 . . 3  |-  ( ( [
C.]  Or  X  /\  X  C_  Fin  /\  A  e.  om )  ->  ( suc  A  e.  om  ->  { b  e.  X  | 
b  ~<_  A }  ~<_  suc  A
) )
587, 57mpd 14 . 2  |-  ( ( [
C.]  Or  X  /\  X  C_  Fin  /\  A  e.  om )  ->  { b  e.  X  |  b  ~<_  A }  ~<_  suc  A
)
59 domfi 7100 . 2  |-  ( ( suc  A  e.  Fin  /\ 
{ b  e.  X  |  b  ~<_  A }  ~<_  suc  A )  ->  { b  e.  X  |  b  ~<_  A }  e.  Fin )
606, 58, 59syl2anc 642 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 176    \/ wo 357    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   {crab 2560    i^i cin 3164    C_ wss 3165   class class class wbr 4039    Or wor 4329   Oncon0 4408   suc csuc 4410   omcom 4672   dom cdm 4705   ` cfv 5271   [ C.] crpss 6292    ~~ cen 6876    ~<_ cdom 6877   Fincfn 6879   cardccrd 7584
This theorem is referenced by:  fin1a2lem11  8052
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-rpss 6293  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-card 7588
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