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Theorem hsmexlem9 8051
Description: Lemma for hsmex 8058. 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 4680 . 2  |-  ( a  e.  om  ->  (
a  =  (/)  \/  E. b  e.  om  a  =  suc  b ) )
2 fveq2 5525 . . . 4  |-  ( a  =  (/)  ->  ( H `
 a )  =  ( H `  (/) ) )
3 hsmexlem7.h . . . . . 6  |-  H  =  ( rec ( ( z  e.  _V  |->  (har
`  ~P ( X  X.  z ) ) ) ,  (har `  ~P X ) )  |`  om )
43hsmexlem7 8049 . . . . 5  |-  ( H `
 (/) )  =  (har
`  ~P X )
5 harcl 7275 . . . . 5  |-  (har `  ~P X )  e.  On
64, 5eqeltri 2353 . . . 4  |-  ( H `
 (/) )  e.  On
72, 6syl6eqel 2371 . . 3  |-  ( a  =  (/)  ->  ( H `
 a )  e.  On )
83hsmexlem8 8050 . . . . . 6  |-  ( b  e.  om  ->  ( H `  suc  b )  =  (har `  ~P ( X  X.  ( H `  b )
) ) )
9 harcl 7275 . . . . . 6  |-  (har `  ~P ( X  X.  ( H `  b )
) )  e.  On
108, 9syl6eqel 2371 . . . . 5  |-  ( b  e.  om  ->  ( H `  suc  b )  e.  On )
11 fveq2 5525 . . . . . 6  |-  ( a  =  suc  b  -> 
( H `  a
)  =  ( H `
 suc  b )
)
1211eleq1d 2349 . . . . 5  |-  ( a  =  suc  b  -> 
( ( H `  a )  e.  On  <->  ( H `  suc  b
)  e.  On ) )
1310, 12syl5ibrcom 213 . . . 4  |-  ( b  e.  om  ->  (
a  =  suc  b  ->  ( H `  a
)  e.  On ) )
1413rexlimiv 2661 . . 3  |-  ( E. b  e.  om  a  =  suc  b  ->  ( H `  a )  e.  On )
157, 14jaoi 368 . 2  |-  ( ( a  =  (/)  \/  E. b  e.  om  a  =  suc  b )  -> 
( H `  a
)  e.  On )
161, 15syl 15 1  |-  ( a  e.  om  ->  ( H `  a )  e.  On )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 357    = wceq 1623    e. wcel 1684   E.wrex 2544   _Vcvv 2788   (/)c0 3455   ~Pcpw 3625    e. cmpt 4077   Oncon0 4392   suc csuc 4394   omcom 4656    X. cxp 4687    |` cres 4691   ` cfv 5255   reccrdg 6422  harchar 7270
This theorem is referenced by:  hsmexlem4  8055  hsmexlem5  8056
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-riota 6304  df-recs 6388  df-rdg 6423  df-en 6864  df-dom 6865  df-oi 7225  df-har 7272
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