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Theorem hsmexlem9 8808
Description: Lemma for hsmex 8815. 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 6709 . 2  |-  ( a  e.  om  ->  (
a  =  (/)  \/  E. b  e.  om  a  =  suc  b ) )
2 fveq2 5856 . . . 4  |-  ( a  =  (/)  ->  ( H `
 a )  =  ( H `  (/) ) )
3 hsmexlem7.h . . . . . 6  |-  H  =  ( rec ( ( z  e.  _V  |->  (har
`  ~P ( X  X.  z ) ) ) ,  (har `  ~P X ) )  |`  om )
43hsmexlem7 8806 . . . . 5  |-  ( H `
 (/) )  =  (har
`  ~P X )
5 harcl 7990 . . . . 5  |-  (har `  ~P X )  e.  On
64, 5eqeltri 2527 . . . 4  |-  ( H `
 (/) )  e.  On
72, 6syl6eqel 2539 . . 3  |-  ( a  =  (/)  ->  ( H `
 a )  e.  On )
83hsmexlem8 8807 . . . . . 6  |-  ( b  e.  om  ->  ( H `  suc  b )  =  (har `  ~P ( X  X.  ( H `  b )
) ) )
9 harcl 7990 . . . . . 6  |-  (har `  ~P ( X  X.  ( H `  b )
) )  e.  On
108, 9syl6eqel 2539 . . . . 5  |-  ( b  e.  om  ->  ( H `  suc  b )  e.  On )
11 fveq2 5856 . . . . . 6  |-  ( a  =  suc  b  -> 
( H `  a
)  =  ( H `
 suc  b )
)
1211eleq1d 2512 . . . . 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 2929 . . 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 1383    e. wcel 1804   E.wrex 2794   _Vcvv 3095   (/)c0 3770   ~Pcpw 3997    |-> cmpt 4495   Oncon0 4868   suc csuc 4870    X. cxp 4987    |` cres 4991   ` cfv 5578   omcom 6685   reccrdg 7077  harchar 7985
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-se 4829  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-isom 5587  df-riota 6242  df-om 6686  df-recs 7044  df-rdg 7078  df-en 7519  df-dom 7520  df-oi 7938  df-har 7987
This theorem is referenced by:  hsmexlem4  8812  hsmexlem5  8813
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