Theorem alephon 3671

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 2832	. . . 4 |- (x = (/) -> (aleph` x) = (aleph` (/)))
2	1	eleq1d 1155	. . 3 |- (x = (/) -> ((aleph` x) e. On <-> (aleph` (/)) e. On))
3		fveq2 2832	. . . 4 |- (x = y -> (aleph` x) = (aleph` y))
4	3	eleq1d 1155	. . 3 |- (x = y -> ((aleph` x) e. On <-> (aleph` y) e. On))
5		fveq2 2832	. . . 4 |- (x = suc y -> (aleph` x) = (aleph` suc y))
6	5	eleq1d 1155	. . 3 |- (x = suc y -> ((aleph` x) e. On <-> (aleph` suc y) e. On))
7		fveq2 2832	. . . 4 |- (x = A -> (aleph` x) = (aleph` A))
8	7	eleq1d 1155	. . 3 |- (x = A -> ((aleph` x) e. On <-> (aleph` A) e. On))
9		aleph0 3669	. . . 4 |- (aleph` (/)) = om
10		omelon 3476	. . . 4 |- om e. On
11	9, 10	eqeltr 1159	. . 3 |- (aleph` (/)) e. On
12		ax-17 925	. . . . . . . . . 10 |- (w e. om -> A.z w e. om)
13		ax-17 925	. . . . . . . . . 10 |- (w e. y -> A.z w e. y)
14		ax-17 925	. . . . . . . . . 10 |- (w e. |^|{x e. On | (aleph` y) ~< x} -> A.z w e. |^|{x e. On | (aleph` y) ~< x})
15		df-aleph 3624	. . . . . . . . . 10 |- aleph = rec({<.z, y>. | y = |^|{x e. On | z ~< x}}, om)
16		breq1 2065	. . . . . . . . . . . 12 |- (z = (aleph` y) -> (z ~< x <-> (aleph` y) ~< x))
17	16	birabsdv 1344	. . . . . . . . . . 11 |- (z = (aleph` y) -> {x e. On | z ~< x} = {x e. On | (aleph` y) ~< x})
18	17	inteqd 1970	. . . . . . . . . 10 |- (z = (aleph` y) -> |^|{x e. On | z ~< x} = |^|{x e. On | (aleph` y) ~< x})
19	12, 13, 14, 15, 18	rdgsucopab 2984	. . . . . . . . 9 |- ((y e. On /\ |^|{x e. On | (aleph` y) ~< x} e. V) -> (aleph` suc y) = |^|{x e. On | (aleph` y) ~< x})
20	19	eleq1d 1155	. . . . . . . 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 2268	. . . . . . . 8 |- (|^|{x e. On | (aleph` y) ~< x} e. V <-> |^|{x e. On | (aleph` y) ~< x} e. On)
22	20, 21	syl6rbbr 417	. . . . . . 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	exp 291	. . . . . 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 451	. . . . 5 |- (y e. On -> (|^|{x e. On | (aleph` y) ~< x} e. V -> (aleph` suc y) e. On))
25		0elon 2277	. . . . . . 7 |- (/) e. On
26	12, 13, 14, 15, 18	rdgsucopabn 2985	. . . . . . . 8 |- (-. |^|{x e. On | (aleph` y) ~< x} e. V -> (aleph` suc y) = (/))
27	26	eleq1d 1155	. . . . . . 7 |- (-. |^|{x e. On | (aleph` y) ~< x} e. V -> ((aleph` suc y) e. On <-> (/) e. On))
28	25, 27	mpbiri 169	. . . . . 6 |- (-. |^|{x e. On | (aleph` y) ~< x} e. V -> (aleph` suc y) e. On)
29	28	a1i 7	. . . . 5 |- (y e. On -> (-. |^|{x e. On | (aleph` y) ~< x} e. V -> (aleph` suc y) e. On))
30	24, 29	pm2.61d 112	. . . 4 |- (y e. On -> (aleph` suc y) e. On)
31	30	a1d 14	. . 3 |- (y e. On -> ((aleph` y) e. On -> (aleph` suc y) e. On))
32		visset 1350	. . . . . 6 |- x e. V
33		alephlim 3670	. . . . . 6 |- ((x e. V /\ Lim x) -> (aleph` x) = U.y e. x (aleph` y))
34	32, 33	mpan 518	. . . . 5 |- (Lim x -> (aleph` x) = U.y e. x (aleph` y))
35	34	eleq1d 1155	. . . 4 |- (Lim x -> ((aleph` x) e. On <-> U.y e. x (aleph` y) e. On))
36		fvex 2838	. . . . 5 |- (aleph` y) e. V
37	32, 36	iunon 2947	. . . 4 |- (A.y e. x (aleph` y) e. On -> U.y e. x (aleph` y) e. On)
38	35, 37	syl5bir 184	. . 3 |- (Lim x -> (A.y e. x (aleph` y) e. On -> (aleph` x) e. On))
39	2, 4, 6, 8, 11, 31, 38	tfinds 2401	. 2 |- (A e. On -> (aleph` A) e. On)
40		alephfnon 3668	. . . . . . . 8 |- aleph Fn On
41		fndm 2723	. . . . . . . 8 |- (aleph Fn On -> dom aleph = On)
42	40, 41	ax-mp 6	. . . . . . 7 |- dom aleph = On
43	42	eleq2i 1153	. . . . . 6 |- (A e. dom aleph <-> A e. On)
44	43	negbii 162	. . . . 5 |- (-. A e. dom aleph <-> -. A e. On)
45		ndmfv 2848	. . . . 5 |- (-. A e. dom aleph -> (aleph` A) = (/))
46	44, 45	sylbir 176	. . . 4 |- (-. A e. On -> (aleph` A) = (/))
47	46	eleq1d 1155	. . 3 |- (-. A e. On -> ((aleph` A) e. On <-> (/) e. On))
48	25, 47	mpbiri 169	. 2 |- (-. A e. On -> (aleph` A) e. On)
49	39, 48	pm2.61i 110	1 |- (aleph` A) e. On

Syntax hints:  -. wn 1   -> wi 2   <-> wb 127   /\ wa 196   = weq 797   e. wel 803   = wceq 1091   e. wcel 1092  A.wral 1201  {crab 1204  Vcvv 1348  (/)c0 1707  |^|cint 1965  U.ciun 1994   class class class wbr 2054  Oncon0 2199  Lim wlim 2200  suc csuc 2201  omcom 2372  dom cdm 2410   Fn wfn 2417  ` cfv 2422   ~< csdm 3273  alephcale 3621

This theorem is referenced by:  alephnbtwn 3674  alephnbtwn2 3675  alephordlem1 3677  alephordlem2 3678  alephordi 3679  alephord 3680  alephord2 3681  alephord3 3683  alephle 3689  cardaleph 3690

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077  ax-reg 1078  ax-inf 1079

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ne 1192  df-ral 1205  df-rex 1206  df-rab 1208  df-v 1349  df-sbc 1441  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-pss 1494  df-nul 1708  df-if 1777  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-int 1966  df-iun 1996  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-lim 2204  df-suc 2205  df-om 2373  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fv 2438  df-rdg 2970  df-aleph 3624

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