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Theorem alephfp2 7738
Description: The aleph function has at least one fixed point. Proposition 11.18 of [TakeutiZaring] p. 104. See alephfp 7737 for an actual example of a fixed point. Compare the inequality alephle 7717 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 7729 . . 3  |-  ran  aleph  C_  On
2 eqid 2285 . . . 4  |-  ( rec ( aleph ,  om )  |` 
om )  =  ( rec ( aleph ,  om )  |`  om )
32alephfplem4 7736 . . 3  |-  U. (
( rec ( aleph ,  om )  |`  om ) " om )  e.  ran  aleph
41, 3sselii 3179 . 2  |-  U. (
( rec ( aleph ,  om )  |`  om ) " om )  e.  On
52alephfp 7737 . 2  |-  ( aleph ` 
U. ( ( rec ( aleph ,  om )  |` 
om ) " om ) )  =  U. ( ( rec ( aleph ,  om )  |`  om ) " om )
6 fveq2 5527 . . . 4  |-  ( x  =  U. ( ( rec ( aleph ,  om )  |`  om ) " om )  ->  ( aleph `  x )  =  (
aleph `  U. ( ( rec ( aleph ,  om )  |`  om ) " om ) ) )
7 id 19 . . . 4  |-  ( x  =  U. ( ( rec ( aleph ,  om )  |`  om ) " om )  ->  x  = 
U. ( ( rec ( aleph ,  om )  |` 
om ) " om ) )
86, 7eqeq12d 2299 . . 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 2886 . 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 653 1  |-  E. x  e.  On  ( aleph `  x
)  =  x
Colors of variables: wff set class
Syntax hints:    = wceq 1625    e. wcel 1686   E.wrex 2546   U.cuni 3829   Oncon0 4394   omcom 4658   ran crn 4692    |` cres 4693   "cima 4694   ` cfv 5257   reccrdg 6424   alephcale 7571
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514  ax-inf2 7344
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 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-reu 2552  df-rmo 2553  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-int 3865  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  df-tr 4116  df-eprel 4307  df-id 4311  df-po 4316  df-so 4317  df-fr 4354  df-se 4355  df-we 4356  df-ord 4397  df-on 4398  df-lim 4399  df-suc 4400  df-om 4659  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-isom 5266  df-riota 6306  df-recs 6390  df-rdg 6425  df-er 6662  df-en 6866  df-dom 6867  df-sdom 6868  df-fin 6869  df-oi 7227  df-har 7274  df-card 7574  df-aleph 7575
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