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Theorem alephfp2 8279
Description: The aleph function has at least one fixed point. Proposition 11.18 of [TakeutiZaring] p. 104. See alephfp 8278 for an actual example of a fixed point. Compare the inequality alephle 8258 that holds in general. Note that if  x is a fixed point, then  aleph `  aleph `  aleph ` ...  aleph `  x  =  x. (Contributed by NM, 6-Nov-2004.) (Revised by Mario Carneiro, 15-May-2015.)
Assertion
Ref Expression
alephfp2  |-  E. x  e.  On  ( aleph `  x
)  =  x

Proof of Theorem alephfp2
StepHypRef Expression
1 alephsson 8270 . . 3  |-  ran  aleph  C_  On
2 eqid 2443 . . . 4  |-  ( rec ( aleph ,  om )  |` 
om )  =  ( rec ( aleph ,  om )  |`  om )
32alephfplem4 8277 . . 3  |-  U. (
( rec ( aleph ,  om )  |`  om ) " om )  e.  ran  aleph
41, 3sselii 3353 . 2  |-  U. (
( rec ( aleph ,  om )  |`  om ) " om )  e.  On
52alephfp 8278 . 2  |-  ( aleph ` 
U. ( ( rec ( aleph ,  om )  |` 
om ) " om ) )  =  U. ( ( rec ( aleph ,  om )  |`  om ) " om )
6 fveq2 5691 . . . 4  |-  ( x  =  U. ( ( rec ( aleph ,  om )  |`  om ) " om )  ->  ( aleph `  x )  =  (
aleph `  U. ( ( rec ( aleph ,  om )  |`  om ) " om ) ) )
7 id 22 . . . 4  |-  ( x  =  U. ( ( rec ( aleph ,  om )  |`  om ) " om )  ->  x  = 
U. ( ( rec ( aleph ,  om )  |` 
om ) " om ) )
86, 7eqeq12d 2457 . . 3  |-  ( x  =  U. ( ( rec ( aleph ,  om )  |`  om ) " om )  ->  ( (
aleph `  x )  =  x  <->  ( aleph `  U. ( ( rec ( aleph ,  om )  |`  om ) " om )
)  =  U. (
( rec ( aleph ,  om )  |`  om ) " om ) ) )
98rspcev 3073 . 2  |-  ( ( U. ( ( rec ( aleph ,  om )  |` 
om ) " om )  e.  On  /\  ( aleph `  U. ( ( rec ( aleph ,  om )  |`  om ) " om ) )  =  U. ( ( rec ( aleph ,  om )  |`  om ) " om )
)  ->  E. x  e.  On  ( aleph `  x
)  =  x )
104, 5, 9mp2an 672 1  |-  E. x  e.  On  ( aleph `  x
)  =  x
Colors of variables: wff setvar class
Syntax hints:    = wceq 1369    e. wcel 1756   E.wrex 2716   U.cuni 4091   Oncon0 4719   ran crn 4841    |` cres 4842   "cima 4843   ` cfv 5418   omcom 6476   reccrdg 6865   alephcale 8106
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-inf2 7847
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-se 4680  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-isom 5427  df-riota 6052  df-om 6477  df-recs 6832  df-rdg 6866  df-er 7101  df-en 7311  df-dom 7312  df-sdom 7313  df-fin 7314  df-oi 7724  df-har 7773  df-card 8109  df-aleph 8110
This theorem is referenced by: (None)
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