Theorem alephfp2 4912

Description: The aleph function has at least one fixed point. Proposition 11.18 of [TakeutiZaring] p. 104. See alephfp 4911 for an actual example of a fixed point. Compare the inequality alephle 4895 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 4905	. . 3 |- ran aleph (_ On
2		eqid 1478	. . . 4 |- (rec({<.y, z>. | z = (aleph` y)}, (aleph` (/))) |` om) = (rec({<.y, z>. | z = (aleph` y)}, (aleph` (/))) |` om)
3	2	alephfplem4 4910	. . 3 |- U.((rec({<.y, z>. | z = (aleph` y)}, (aleph` (/))) |` om)"om) e. ran aleph
4	1, 3	sselii 2069	. 2 |- U.((rec({<.y, z>. | z = (aleph` y)}, (aleph` (/))) |` om)"om) e. On
5	2	alephfp 4911	. 2 |- (aleph` U.((rec({<.y, z>. | z = (aleph` y)}, (aleph` (/))) |` om)"om)) = U.((rec({<.y, z>. | z = (aleph` y)}, (aleph` (/))) |` om)"om)
6		fveq2 3730	. . . 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 59	. . . 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 1492	. . 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 1880	. 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 699	1 |- E.x e. On (aleph` x) = x

Syntax hints:   = wceq 958   e. wcel 960  E.wrex 1649  (/)c0 2283  U.cuni 2507  {copab 2671  Oncon0 2954  omcom 3137  ran crn 3177   |` cres 3178  "cima 3179  ` cfv 3188  reccrdg 3937  alephcale 4824

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-rep 2698  ax-sep 2708  ax-nul 2715  ax-pow 2748  ax-pr 2785  ax-un 2872  ax-reg 4602  ax-inf2 4634  ax-ac 4754

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 778  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-reu 1654  df-rab 1655  df-v 1815  df-sbc 1945  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-pss 2058  df-nul 2284  df-if 2366  df-pw 2406  df-sn 2416  df-pr 2417  df-tp 2419  df-op 2420  df-uni 2508  df-int 2538  df-iun 2572  df-br 2625  df-opab 2672  df-tr 2686  df-eprel 2838  df-id 2841  df-po 2846  df-so 2856  df-fr 2923  df-we 2940  df-ord 2957  df-on 2958  df-lim 2959  df-suc 2960  df-om 3138  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-f 3200  df-f1 3201  df-fo 3202  df-f1o 3203  df-fv 3204  df-rdg 3938  df-er 4267  df-en 4374  df-dom 4375  df-sdom 4376  df-fin 4377  df-card 4826  df-aleph 4827

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