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Theorem hsmexlem9 8697
Description: Lemma for hsmex 8704. 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 6602 . 2  |-  ( a  e.  om  ->  (
a  =  (/)  \/  E. b  e.  om  a  =  suc  b ) )
2 fveq2 5791 . . . 4  |-  ( a  =  (/)  ->  ( H `
 a )  =  ( H `  (/) ) )
3 hsmexlem7.h . . . . . 6  |-  H  =  ( rec ( ( z  e.  _V  |->  (har
`  ~P ( X  X.  z ) ) ) ,  (har `  ~P X ) )  |`  om )
43hsmexlem7 8695 . . . . 5  |-  ( H `
 (/) )  =  (har
`  ~P X )
5 harcl 7879 . . . . 5  |-  (har `  ~P X )  e.  On
64, 5eqeltri 2535 . . . 4  |-  ( H `
 (/) )  e.  On
72, 6syl6eqel 2547 . . 3  |-  ( a  =  (/)  ->  ( H `
 a )  e.  On )
83hsmexlem8 8696 . . . . . 6  |-  ( b  e.  om  ->  ( H `  suc  b )  =  (har `  ~P ( X  X.  ( H `  b )
) ) )
9 harcl 7879 . . . . . 6  |-  (har `  ~P ( X  X.  ( H `  b )
) )  e.  On
108, 9syl6eqel 2547 . . . . 5  |-  ( b  e.  om  ->  ( H `  suc  b )  e.  On )
11 fveq2 5791 . . . . . 6  |-  ( a  =  suc  b  -> 
( H `  a
)  =  ( H `
 suc  b )
)
1211eleq1d 2520 . . . . 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 2933 . . 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 1370    e. wcel 1758   E.wrex 2796   _Vcvv 3070   (/)c0 3737   ~Pcpw 3960    |-> cmpt 4450   Oncon0 4819   suc csuc 4821    X. cxp 4938    |` cres 4942   ` cfv 5518   omcom 6578   reccrdg 6967  harchar 7874
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 1952  ax-ext 2430  ax-rep 4503  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474
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 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-pss 3444  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-tp 3982  df-op 3984  df-uni 4192  df-iun 4273  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4486  df-eprel 4732  df-id 4736  df-po 4741  df-so 4742  df-fr 4779  df-se 4780  df-we 4781  df-ord 4822  df-on 4823  df-lim 4824  df-suc 4825  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-isom 5527  df-riota 6153  df-om 6579  df-recs 6934  df-rdg 6968  df-en 7413  df-dom 7414  df-oi 7827  df-har 7876
This theorem is referenced by:  hsmexlem4  8701  hsmexlem5  8702
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