Theorem alephfp2 6049

Description: The aleph function has at least one fixed point. Proposition 11.18 of [TakeutiZaring] p. 104. See alephfp 6048 for an actual example of a fixed point. Compare the inequality alephle 6032 that holds in general. Note that if x is a fixed point, then aleph` aleph` aleph` ... aleph` x = x.

Assertion

Ref	Expression
alephfp2	|- E.x e. On (aleph` x) = x

Proof of Theorem alephfp2

Step	Hyp	Ref	Expression
1		alephsson 6042	. . 3 |- ran aleph C_ On
2		eqid 1884	. . . 4 |- (rec({<.y, z>. | z = (aleph` y)}, (aleph` (/))) |` om) = (rec({<.y, z>. | z = (aleph` y)}, (aleph` (/))) |` om)
3	2	alephfplem4 6047	. . 3 |- U.((rec({<.y, z>. | z = (aleph` y)}, (aleph` (/))) |` om)"om) e. ran aleph
4	1, 3	sselii 2618	. 2 |- U.((rec({<.y, z>. | z = (aleph` y)}, (aleph` (/))) |` om)"om) e. On
5	2	alephfp 6048	. 2 |- (aleph` U.((rec({<.y, z>. | z = (aleph` y)}, (aleph` (/))) |` om)"om)) = U.((rec({<.y, z>. | z = (aleph` y)}, (aleph` (/))) |` om)"om)
6		fveq2 4681	. . . 4 |- (x = U.((rec({<.y, z>. | z = (aleph` y)}, (aleph` (/))) |` om)"om) -> (aleph` x) = (aleph` U.((rec({<.y, z>. | z = (aleph` y)}, (aleph` (/))) |` om)"om)))
7		id 73	. . . 4 |- (x = U.((rec({<.y, z>. | z = (aleph` y)}, (aleph` (/))) |` om)"om) -> x = U.((rec({<.y, z>. | z = (aleph` y)}, (aleph` (/))) |` om)"om))
8	6, 7	eqeq12d 1899	. . 3 |- (x = U.((rec({<.y, z>. | z = (aleph` y)}, (aleph` (/))) |` om)"om) -> ((aleph` x) = x <-> (aleph` U.((rec({<.y, z>. | z = (aleph` y)}, (aleph` (/))) |` om)"om)) = U.((rec({<.y, z>. | z = (aleph` y)}, (aleph` (/))) |` om)"om)))
9	8	rcla4ev 2381	. 2 |- ((U.((rec({<.y, z>. | z = (aleph` y)}, (aleph` (/))) |` om)"om) e. On /\ (aleph` U.((rec({<.y, z>. | z = (aleph` y)}, (aleph` (/))) |` om)"om)) = U.((rec({<.y, z>. | z = (aleph` y)}, (aleph` (/))) |` om)"om)) -> E.x e. On (aleph` x) = x)
10	4, 5, 9	mp2an 761	1 |- E.x e. On (aleph` x) = x

Syntax hints:   = wceq 1298   e. wcel 1300  E.wrex 2106  (/)c0 2875  U.cuni 3177  {copab 3395  Oncon0 3657  omcom 3949  ran crn 3987   |` cres 3988  "cima 3989  ` cfv 3998  reccrdg 5139  alephcale 5860

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-reg 5695  ax-inf2 5731  ax-ac 5906

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-rdg 5140  df-er 5318  df-en 5427  df-dom 5428  df-sdom 5429  df-fin 5430  df-card 5862  df-aleph 5863

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