Theorem alephon 4837

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 3709	. . . 4 |- (x = (/) -> (aleph` x) = (aleph` (/)))
2	1	eleq1d 1532	. . 3 |- (x = (/) -> ((aleph` x) e. On <-> (aleph` (/)) e. On))
3		fveq2 3709	. . . 4 |- (x = y -> (aleph` x) = (aleph` y))
4	3	eleq1d 1532	. . 3 |- (x = y -> ((aleph` x) e. On <-> (aleph` y) e. On))
5		fveq2 3709	. . . 4 |- (x = suc y -> (aleph` x) = (aleph` suc y))
6	5	eleq1d 1532	. . 3 |- (x = suc y -> ((aleph` x) e. On <-> (aleph` suc y) e. On))
7		fveq2 3709	. . . 4 |- (x = A -> (aleph` x) = (aleph` A))
8	7	eleq1d 1532	. . 3 |- (x = A -> ((aleph` x) e. On <-> (aleph` A) e. On))
9		aleph0 4835	. . . 4 |- (aleph` (/)) = om
10		omelon 4601	. . . 4 |- om e. On
11	9, 10	eqeltr 1536	. . 3 |- (aleph` (/)) e. On
12		ax-17 968	. . . . . . . . . 10 |- (w e. om -> A.z w e. om)
13		ax-17 968	. . . . . . . . . 10 |- (w e. y -> A.z w e. y)
14		ax-17 968	. . . . . . . . . 10 |- (w e. |^|{x e. On | (aleph` y) ~< x} -> A.z w e. |^|{x e. On | (aleph` y) ~< x})
15		df-aleph 4789	. . . . . . . . . 10 |- aleph = rec({<.z, y>. | y = |^|{x e. On | z ~< x}}, om)
16		breq1 2612	. . . . . . . . . . . 12 |- (z = (aleph` y) -> (z ~< x <-> (aleph` y) ~< x))
17	16	rabbisdv 1798	. . . . . . . . . . 11 |- (z = (aleph` y) -> {x e. On | z ~< x} = {x e. On | (aleph` y) ~< x})
18	17	inteqd 2528	. . . . . . . . . 10 |- (z = (aleph` y) -> |^|{x e. On | z ~< x} = |^|{x e. On | (aleph` y) ~< x})
19	12, 13, 14, 15, 18	rdgsucopab 3931	. . . . . . . . 9 |- ((y e. On /\ |^|{x e. On | (aleph` y) ~< x} e. V) -> (aleph` suc y) = |^|{x e. On | (aleph` y) ~< x})
20	19	eleq1d 1532	. . . . . . . 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 3003	. . . . . . . 8 |- (|^|{x e. On | (aleph` y) ~< x} e. V <-> |^|{x e. On | (aleph` y) ~< x} e. On)
22	20, 21	syl6rbbr 537	. . . . . . 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 373	. . . . . 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 592	. . . . 5 |- (y e. On -> (|^|{x e. On | (aleph` y) ~< x} e. V -> (aleph` suc y) e. On))
25	12, 13, 14, 15, 18	rdgsucopabn 3932	. . . . . 6 |- (-. |^|{x e. On | (aleph` y) ~< x} e. V -> (aleph` suc y) = (/))
26		0elon 3012	. . . . . 6 |- (/) e. On
27	25, 26	syl6eqel 1548	. . . . 5 |- (-. |^|{x e. On | (aleph` y) ~< x} e. V -> (aleph` suc y) e. On)
28	24, 27	pm2.61d1 128	. . . 4 |- (y e. On -> (aleph` suc y) e. On)
29	28	a1d 12	. . 3 |- (y e. On -> ((aleph` y) e. On -> (aleph` suc y) e. On))
30		visset 1804	. . . . . 6 |- x e. V
31		alephlim 4836	. . . . . 6 |- ((x e. V /\ Lim x) -> (aleph` x) = U_y e. x (aleph` y))
32	30, 31	mpan 693	. . . . 5 |- (Lim x -> (aleph` x) = U_y e. x (aleph` y))
33	32	eleq1d 1532	. . . 4 |- (Lim x -> ((aleph` x) e. On <-> U_y e. x (aleph` y) e. On))
34		fvex 3717	. . . . 5 |- (aleph` y) e. V
35	30, 34	iunon 3894	. . . 4 |- (A.y e. x (aleph` y) e. On -> U_y e. x (aleph` y) e. On)
36	33, 35	syl5bir 210	. . 3 |- (Lim x -> (A.y e. x (aleph` y) e. On -> (aleph` x) e. On))
37	2, 4, 6, 8, 11, 29, 36	tfinds 3151	. 2 |- (A e. On -> (aleph` A) e. On)
38		alephfnon 4834	. . . . . . 7 |- aleph Fn On
39		fndm 3573	. . . . . . 7 |- (aleph Fn On -> dom aleph = On)
40	38, 39	ax-mp 7	. . . . . 6 |- dom aleph = On
41	40	eleq2i 1530	. . . . 5 |- (A e. dom aleph <-> A e. On)
42	41	negbii 187	. . . 4 |- (-. A e. dom aleph <-> -. A e. On)
43		ndmfv 3730	. . . 4 |- (-. A e. dom aleph -> (aleph` A) = (/))
44	42, 43	sylbir 201	. . 3 |- (-. A e. On -> (aleph` A) = (/))
45	44, 26	syl6eqel 1548	. 2 |- (-. A e. On -> (aleph` A) e. On)
46	37, 45	pm2.61i 126	1 |- (aleph` A) e. On

Syntax hints:  -. wn 2   <-> wb 146   /\ wa 223   = wceq 953   e. wcel 955  A.wral 1637  {crab 1640  Vcvv 1802  (/)c0 2270  |^|cint 2523  U_ciun 2556   class class class wbr 2609  Oncon0 2938  Lim wlim 2939  suc csuc 2940  omcom 3121  dom cdm 3160   Fn wfn 3167  ` cfv 3172   ~< csdm 4350  alephcale 4786

This theorem is referenced by:  alephnbtwn 4840  alephnbtwn2 4841  alephordlem1 4844  alephordlem2 4845  alephordi 4846  alephord 4847  alephord2 4848  alephord3 4850  alephle 4856  cardaleph 4857  alephfp 4872  alephval2 4874

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-rep 2683  ax-sep 2693  ax-nul 2700  ax-pow 2732  ax-pr 2769  ax-un 2857  ax-inf2 4597

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 774  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-rab 1644  df-v 1803  df-sbc 1932  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-if 2352  df-pw 2392  df-sn 2402  df-pr 2403  df-tp 2405  df-op 2406  df-uni 2494  df-int 2524  df-iun 2558  df-br 2610  df-opab 2657  df-tr 2671  df-eprel 2821  df-id 2824  df-po 2831  df-so 2841  df-fr 2907  df-we 2924  df-ord 2941  df-on 2942  df-lim 2943  df-suc 2944  df-om 3122  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-fv 3188  df-rdg 3917  df-aleph 4789

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