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Theorem alephfp2 7732
Description: The aleph function has at least one fixed point. Proposition 11.18 of [TakeutiZaring] p. 104. See alephfp 7731 for an actual example of a fixed point. Compare the inequality alephle 7711 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 7723 . . 3  |-  ran  aleph  C_  On
2 eqid 2284 . . . 4  |-  ( rec ( aleph ,  om )  |` 
om )  =  ( rec ( aleph ,  om )  |`  om )
32alephfplem4 7730 . . 3  |-  U. (
( rec ( aleph ,  om )  |`  om ) " om )  e.  ran  aleph
41, 3sselii 3178 . 2  |-  U. (
( rec ( aleph ,  om )  |`  om ) " om )  e.  On
52alephfp 7731 . 2  |-  ( aleph ` 
U. ( ( rec ( aleph ,  om )  |` 
om ) " om ) )  =  U. ( ( rec ( aleph ,  om )  |`  om ) " om )
6 fveq2 5486 . . . 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 2298 . . 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 2885 . 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 1623    e. wcel 1685   E.wrex 2545   U.cuni 3828   Oncon0 4391   omcom 4655   ran crn 4689    |` cres 4690   "cima 4691   ` cfv 5221   reccrdg 6418   alephcale 7565
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 1636  ax-8 1644  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511  ax-inf2 7338
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 1631  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-reu 2551  df-rmo 2552  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-int 3864  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-se 4352  df-we 4353  df-ord 4394  df-on 4395  df-lim 4396  df-suc 4397  df-om 4656  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-isom 5230  df-iota 6253  df-riota 6300  df-recs 6384  df-rdg 6419  df-er 6656  df-en 6860  df-dom 6861  df-sdom 6862  df-fin 6863  df-oi 7221  df-har 7268  df-card 7568  df-aleph 7569
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