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Theorem alephfp2 7736
Description: The aleph function has at least one fixed point. Proposition 11.18 of [TakeutiZaring] p. 104. See alephfp 7735 for an actual example of a fixed point. Compare the inequality alephle 7715 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 7727 . . 3  |-  ran  aleph  C_  On
2 eqid 2283 . . . 4  |-  ( rec ( aleph ,  om )  |` 
om )  =  ( rec ( aleph ,  om )  |`  om )
32alephfplem4 7734 . . 3  |-  U. (
( rec ( aleph ,  om )  |`  om ) " om )  e.  ran  aleph
41, 3sselii 3177 . 2  |-  U. (
( rec ( aleph ,  om )  |`  om ) " om )  e.  On
52alephfp 7735 . 2  |-  ( aleph ` 
U. ( ( rec ( aleph ,  om )  |` 
om ) " om ) )  =  U. ( ( rec ( aleph ,  om )  |`  om ) " om )
6 fveq2 5525 . . . 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 2297 . . 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 2884 . 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 1684   E.wrex 2544   U.cuni 3827   Oncon0 4392   omcom 4656   ran crn 4690    |` cres 4691   "cima 4692   ` cfv 5255   reccrdg 6422   alephcale 7569
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  ax-inf2 7342
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-int 3863  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-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-oi 7225  df-har 7272  df-card 7572  df-aleph 7573
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