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Theorem hsmexlem9 8801
Description: Lemma for hsmex 8808. Properties of the recurrent sequence of ordinals. (Contributed by Stefan O'Rear, 14-Feb-2015.)
Hypothesis
Ref Expression
hsmexlem7.h  |-  H  =  ( rec ( ( z  e.  _V  |->  (har
`  ~P ( X  X.  z ) ) ) ,  (har `  ~P X ) )  |`  om )
Assertion
Ref Expression
hsmexlem9  |-  ( a  e.  om  ->  ( H `  a )  e.  On )
Distinct variable groups:    z, X    z, a
Allowed substitution hints:    H( z, a)    X( a)

Proof of Theorem hsmexlem9
Dummy variable  b is distinct from all other variables.
StepHypRef Expression
1 nn0suc 6702 . 2  |-  ( a  e.  om  ->  (
a  =  (/)  \/  E. b  e.  om  a  =  suc  b ) )
2 fveq2 5864 . . . 4  |-  ( a  =  (/)  ->  ( H `
 a )  =  ( H `  (/) ) )
3 hsmexlem7.h . . . . . 6  |-  H  =  ( rec ( ( z  e.  _V  |->  (har
`  ~P ( X  X.  z ) ) ) ,  (har `  ~P X ) )  |`  om )
43hsmexlem7 8799 . . . . 5  |-  ( H `
 (/) )  =  (har
`  ~P X )
5 harcl 7983 . . . . 5  |-  (har `  ~P X )  e.  On
64, 5eqeltri 2551 . . . 4  |-  ( H `
 (/) )  e.  On
72, 6syl6eqel 2563 . . 3  |-  ( a  =  (/)  ->  ( H `
 a )  e.  On )
83hsmexlem8 8800 . . . . . 6  |-  ( b  e.  om  ->  ( H `  suc  b )  =  (har `  ~P ( X  X.  ( H `  b )
) ) )
9 harcl 7983 . . . . . 6  |-  (har `  ~P ( X  X.  ( H `  b )
) )  e.  On
108, 9syl6eqel 2563 . . . . 5  |-  ( b  e.  om  ->  ( H `  suc  b )  e.  On )
11 fveq2 5864 . . . . . 6  |-  ( a  =  suc  b  -> 
( H `  a
)  =  ( H `
 suc  b )
)
1211eleq1d 2536 . . . . 5  |-  ( a  =  suc  b  -> 
( ( H `  a )  e.  On  <->  ( H `  suc  b
)  e.  On ) )
1310, 12syl5ibrcom 222 . . . 4  |-  ( b  e.  om  ->  (
a  =  suc  b  ->  ( H `  a
)  e.  On ) )
1413rexlimiv 2949 . . 3  |-  ( E. b  e.  om  a  =  suc  b  ->  ( H `  a )  e.  On )
157, 14jaoi 379 . 2  |-  ( ( a  =  (/)  \/  E. b  e.  om  a  =  suc  b )  -> 
( H `  a
)  e.  On )
161, 15syl 16 1  |-  ( a  e.  om  ->  ( H `  a )  e.  On )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 368    = wceq 1379    e. wcel 1767   E.wrex 2815   _Vcvv 3113   (/)c0 3785   ~Pcpw 4010    |-> cmpt 4505   Oncon0 4878   suc csuc 4880    X. cxp 4997    |` cres 5001   ` cfv 5586   omcom 6678   reccrdg 7072  harchar 7978
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-isom 5595  df-riota 6243  df-om 6679  df-recs 7039  df-rdg 7073  df-en 7514  df-dom 7515  df-oi 7931  df-har 7980
This theorem is referenced by:  hsmexlem4  8805  hsmexlem5  8806
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