Theorem alephon 5876

Description: An aleph is an ordinal number.

Assertion

Ref	Expression
alephon	|- (aleph` A) e. On

Proof of Theorem alephon

Step	Hyp	Ref	Expression
1		fveq2 4681	. . . 4 |- (x = (/) -> (aleph` x) = (aleph` (/)))
2	1	eleq1d 1963	. . 3 |- (x = (/) -> ((aleph` x) e. On <-> (aleph` (/)) e. On))
3		fveq2 4681	. . . 4 |- (x = y -> (aleph` x) = (aleph` y))
4	3	eleq1d 1963	. . 3 |- (x = y -> ((aleph` x) e. On <-> (aleph` y) e. On))
5		fveq2 4681	. . . 4 |- (x = suc y -> (aleph` x) = (aleph` suc y))
6	5	eleq1d 1963	. . 3 |- (x = suc y -> ((aleph` x) e. On <-> (aleph` suc y) e. On))
7		fveq2 4681	. . . 4 |- (x = A -> (aleph` x) = (aleph` A))
8	7	eleq1d 1963	. . 3 |- (x = A -> ((aleph` x) e. On <-> (aleph` A) e. On))
9		aleph0 5874	. . . 4 |- (aleph` (/)) = om
10		omelon 5736	. . . 4 |- om e. On
11	9, 10	eqeltri 1967	. . 3 |- (aleph` (/)) e. On
12		ax-17 1317	. . . . . . . . . 10 |- (w e. om -> A.z w e. om)
13		ax-17 1317	. . . . . . . . . 10 |- (w e. y -> A.z w e. y)
14		ax-17 1317	. . . . . . . . . 10 |- (w e. |^|{x e. On | (aleph` y) ~< x} -> A.z w e. |^|{x e. On | (aleph` y) ~< x})
15		df-aleph 5863	. . . . . . . . . 10 |- aleph = rec({<.z, y>. | y = |^|{x e. On | z ~< x}}, om)
16		breq1 3341	. . . . . . . . . . . 12 |- (z = (aleph` y) -> (z ~< x <-> (aleph` y) ~< x))
17	16	rabbidv 2287	. . . . . . . . . . 11 |- (z = (aleph` y) -> {x e. On | z ~< x} = {x e. On | (aleph` y) ~< x})
18	17	inteqd 3219	. . . . . . . . . 10 |- (z = (aleph` y) -> |^|{x e. On | z ~< x} = |^|{x e. On | (aleph` y) ~< x})
19	12, 13, 14, 15, 18	rdgsucopab 5154	. . . . . . . . 9 |- ((y e. On /\ |^|{x e. On | (aleph` y) ~< x} e. _V) -> (aleph` suc y) = |^|{x e. On | (aleph` y) ~< x})
20	19	eleq1d 1963	. . . . . . . 8 |- ((y e. On /\ |^|{x e. On | (aleph` y) ~< x} e. _V) -> ((aleph` suc y) e. On <-> |^|{x e. On | (aleph` y) ~< x} e. On))
21		onintrab 3882	. . . . . . . 8 |- (|^|{x e. On | (aleph` y) ~< x} e. _V <-> |^|{x e. On | (aleph` y) ~< x} e. On)
22	20, 21	syl6rbbr 598	. . . . . . 7 |- ((y e. On /\ |^|{x e. On | (aleph` y) ~< x} e. _V) -> (|^|{x e. On | (aleph` y) ~< x} e. _V <-> (aleph` suc y) e. On))
23	22	ex 402	. . . . . 6 |- (y e. On -> (|^|{x e. On | (aleph` y) ~< x} e. _V -> (|^|{x e. On | (aleph` y) ~< x} e. _V <-> (aleph` suc y) e. On)))
24	23	ibd 654	. . . . 5 |- (y e. On -> (|^|{x e. On | (aleph` y) ~< x} e. _V -> (aleph` suc y) e. On))
25	12, 13, 14, 15, 18	rdgsucopabn 5155	. . . . . 6 |- (-. |^|{x e. On | (aleph` y) ~< x} e. _V -> (aleph` suc y) = (/))
26		0elon 3716	. . . . . 6 |- (/) e. On
27	25, 26	syl6eqel 1979	. . . . 5 |- (-. |^|{x e. On | (aleph` y) ~< x} e. _V -> (aleph` suc y) e. On)
28	24, 27	pm2.61d1 142	. . . 4 |- (y e. On -> (aleph` suc y) e. On)
29	28	a1d 15	. . 3 |- (y e. On -> ((aleph` y) e. On -> (aleph` suc y) e. On))
30		visset 2295	. . . . . 6 |- x e. _V
31		alephlim 5875	. . . . . 6 |- ((x e. _V /\ Lim x) -> (aleph` x) = U_y e. x (aleph` y))
32	30, 31	mpan 759	. . . . 5 |- (Lim x -> (aleph` x) = U_y e. x (aleph` y))
33	32	eleq1d 1963	. . . 4 |- (Lim x -> ((aleph` x) e. On <-> U_y e. x (aleph` y) e. On))
34		fvex 4689	. . . . 5 |- (aleph` y) e. _V
35	30, 34	iunon 5114	. . . 4 |- (A.y e. x (aleph` y) e. On -> U_y e. x (aleph` y) e. On)
36	33, 35	syl5bir 227	. . 3 |- (Lim x -> (A.y e. x (aleph` y) e. On -> (aleph` x) e. On))
37	2, 4, 6, 8, 11, 29, 36	tfinds 3942	. 2 |- (A e. On -> (aleph` A) e. On)
38		alephfnon 5873	. . . . . . 7 |- aleph Fn On
39		fndm 4512	. . . . . . 7 |- (aleph Fn On -> dom aleph = On)
40	38, 39	ax-mp 7	. . . . . 6 |- dom aleph = On
41	40	eleq2i 1961	. . . . 5 |- (A e. dom aleph <-> A e. On)
42	41	notbii 204	. . . 4 |- (-. A e. dom aleph <-> -. A e. On)
43		ndmfv 4702	. . . 4 |- (-. A e. dom aleph -> (aleph` A) = (/))
44	42, 43	sylbir 218	. . 3 |- (-. A e. On -> (aleph` A) = (/))
45	44, 26	syl6eqel 1979	. 2 |- (-. A e. On -> (aleph` A) e. On)
46	37, 45	pm2.61i 140	1 |- (aleph` A) e. On

Syntax hints:  -. wn 2   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  A.wral 2105  {crab 2108  _Vcvv 2292  (/)c0 2875  |^|cint 3214  U_ciun 3255   class class class wbr 3338  Oncon0 3657  Lim wlim 3658  suc csuc 3659  omcom 3949  dom cdm 3986   Fn wfn 3993  ` cfv 3998   ~< csdm 5425  alephcale 5860

This theorem is referenced by:  omsubsuc 5877  omsubsuc2 5878  omsubsdomlem1 5879  omsubel 5883  omsubss 5884  elomsubsd 5885  omsubdmss 5886  omsublim 5887  omsubindss 5888  infenomsub 5889  omsubinit 5890  alephnbtwn 6016  alephnbtwn2 6017  alephordlem1 6020  alephordlem2 6021  alephordi 6022  alephord 6023  alephord2 6024  alephord3 6026  alephle 6032  cardaleph 6033  alephfp 6048  alephval2 6050  omsubsucOLD 15386  omsubsuc2OLD 15387  omsubsdomlem1OLD 15388  omsubelOLD 15392  omsubssOLD 15393  elomsubsdOLD 15394  omsubdmssOLD 15395  omsublimOLD 15396  omsubindssOLD 15397  infenomsubOLD 15398  omsubinitOLD 15399

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-inf2 5731

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-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-fv 4014  df-rdg 5140  df-aleph 5863

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