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Theorem alephfp2 7983
Description: The aleph function has at least one fixed point. Proposition 11.18 of [TakeutiZaring] p. 104. See alephfp 7982 for an actual example of a fixed point. Compare the inequality alephle 7962 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 7974 . . 3  |-  ran  aleph  C_  On
2 eqid 2436 . . . 4  |-  ( rec ( aleph ,  om )  |` 
om )  =  ( rec ( aleph ,  om )  |`  om )
32alephfplem4 7981 . . 3  |-  U. (
( rec ( aleph ,  om )  |`  om ) " om )  e.  ran  aleph
41, 3sselii 3338 . 2  |-  U. (
( rec ( aleph ,  om )  |`  om ) " om )  e.  On
52alephfp 7982 . 2  |-  ( aleph ` 
U. ( ( rec ( aleph ,  om )  |` 
om ) " om ) )  =  U. ( ( rec ( aleph ,  om )  |`  om ) " om )
6 fveq2 5721 . . . 4  |-  ( x  =  U. ( ( rec ( aleph ,  om )  |`  om ) " om )  ->  ( aleph `  x )  =  (
aleph `  U. ( ( rec ( aleph ,  om )  |`  om ) " om ) ) )
7 id 20 . . . 4  |-  ( x  =  U. ( ( rec ( aleph ,  om )  |`  om ) " om )  ->  x  = 
U. ( ( rec ( aleph ,  om )  |` 
om ) " om ) )
86, 7eqeq12d 2450 . . 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 3045 . 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 654 1  |-  E. x  e.  On  ( aleph `  x
)  =  x
Colors of variables: wff set class
Syntax hints:    = wceq 1652    e. wcel 1725   E.wrex 2699   U.cuni 4008   Oncon0 4574   omcom 4838   ran crn 4872    |` cres 4873   "cima 4874   ` cfv 5447   reccrdg 6660   alephcale 7816
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4313  ax-sep 4323  ax-nul 4331  ax-pow 4370  ax-pr 4396  ax-un 4694  ax-inf2 7589
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2703  df-rex 2704  df-reu 2705  df-rmo 2706  df-rab 2707  df-v 2951  df-sbc 3155  df-csb 3245  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-pss 3329  df-nul 3622  df-if 3733  df-pw 3794  df-sn 3813  df-pr 3814  df-tp 3815  df-op 3816  df-uni 4009  df-int 4044  df-iun 4088  df-br 4206  df-opab 4260  df-mpt 4261  df-tr 4296  df-eprel 4487  df-id 4491  df-po 4496  df-so 4497  df-fr 4534  df-se 4535  df-we 4536  df-ord 4577  df-on 4578  df-lim 4579  df-suc 4580  df-om 4839  df-xp 4877  df-rel 4878  df-cnv 4879  df-co 4880  df-dm 4881  df-rn 4882  df-res 4883  df-ima 4884  df-iota 5411  df-fun 5449  df-fn 5450  df-f 5451  df-f1 5452  df-fo 5453  df-f1o 5454  df-fv 5455  df-isom 5456  df-riota 6542  df-recs 6626  df-rdg 6661  df-er 6898  df-en 7103  df-dom 7104  df-sdom 7105  df-fin 7106  df-oi 7472  df-har 7519  df-card 7819  df-aleph 7820
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