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Theorem hsmexlem9 8067
Description: Lemma for hsmex 8074. 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 4696 . 2  |-  ( a  e.  om  ->  (
a  =  (/)  \/  E. b  e.  om  a  =  suc  b ) )
2 fveq2 5541 . . . 4  |-  ( a  =  (/)  ->  ( H `
 a )  =  ( H `  (/) ) )
3 hsmexlem7.h . . . . . 6  |-  H  =  ( rec ( ( z  e.  _V  |->  (har
`  ~P ( X  X.  z ) ) ) ,  (har `  ~P X ) )  |`  om )
43hsmexlem7 8065 . . . . 5  |-  ( H `
 (/) )  =  (har
`  ~P X )
5 harcl 7291 . . . . 5  |-  (har `  ~P X )  e.  On
64, 5eqeltri 2366 . . . 4  |-  ( H `
 (/) )  e.  On
72, 6syl6eqel 2384 . . 3  |-  ( a  =  (/)  ->  ( H `
 a )  e.  On )
83hsmexlem8 8066 . . . . . 6  |-  ( b  e.  om  ->  ( H `  suc  b )  =  (har `  ~P ( X  X.  ( H `  b )
) ) )
9 harcl 7291 . . . . . 6  |-  (har `  ~P ( X  X.  ( H `  b )
) )  e.  On
108, 9syl6eqel 2384 . . . . 5  |-  ( b  e.  om  ->  ( H `  suc  b )  e.  On )
11 fveq2 5541 . . . . . 6  |-  ( a  =  suc  b  -> 
( H `  a
)  =  ( H `
 suc  b )
)
1211eleq1d 2362 . . . . 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 2674 . . 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 1632    e. wcel 1696   E.wrex 2557   _Vcvv 2801   (/)c0 3468   ~Pcpw 3638    e. cmpt 4093   Oncon0 4408   suc csuc 4410   omcom 4672    X. cxp 4703    |` cres 4707   ` cfv 5271   reccrdg 6438  harchar 7286
This theorem is referenced by:  hsmexlem4  8071  hsmexlem5  8072
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-rmo 2564  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-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-se 4369  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-isom 5280  df-riota 6320  df-recs 6404  df-rdg 6439  df-en 6880  df-dom 6881  df-oi 7241  df-har 7288
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