Theorem alephon 4876

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 3730	. . . 4 |- (x = (/) -> (aleph` x) = (aleph` (/)))
2	1	eleq1d 1543	. . 3 |- (x = (/) -> ((aleph` x) e. On <-> (aleph` (/)) e. On))
3		fveq2 3730	. . . 4 |- (x = y -> (aleph` x) = (aleph` y))
4	3	eleq1d 1543	. . 3 |- (x = y -> ((aleph` x) e. On <-> (aleph` y) e. On))
5		fveq2 3730	. . . 4 |- (x = suc y -> (aleph` x) = (aleph` suc y))
6	5	eleq1d 1543	. . 3 |- (x = suc y -> ((aleph` x) e. On <-> (aleph` suc y) e. On))
7		fveq2 3730	. . . 4 |- (x = A -> (aleph` x) = (aleph` A))
8	7	eleq1d 1543	. . 3 |- (x = A -> ((aleph` x) e. On <-> (aleph` A) e. On))
9		aleph0 4874	. . . 4 |- (aleph` (/)) = om
10		omelon 4638	. . . 4 |- om e. On
11	9, 10	eqeltr 1547	. . 3 |- (aleph` (/)) e. On
12		ax-17 973	. . . . . . . . . 10 |- (w e. om -> A.z w e. om)
13		ax-17 973	. . . . . . . . . 10 |- (w e. y -> A.z w e. y)
14		ax-17 973	. . . . . . . . . 10 |- (w e. |^|{x e. On | (aleph` y) ~< x} -> A.z w e. |^|{x e. On | (aleph` y) ~< x})
15		df-aleph 4827	. . . . . . . . . 10 |- aleph = rec({<.z, y>. | y = |^|{x e. On | z ~< x}}, om)
16		breq1 2627	. . . . . . . . . . . 12 |- (z = (aleph` y) -> (z ~< x <-> (aleph` y) ~< x))
17	16	rabbisdv 1810	. . . . . . . . . . 11 |- (z = (aleph` y) -> {x e. On | z ~< x} = {x e. On | (aleph` y) ~< x})
18	17	inteqd 2542	. . . . . . . . . 10 |- (z = (aleph` y) -> |^|{x e. On | z ~< x} = |^|{x e. On | (aleph` y) ~< x})
19	12, 13, 14, 15, 18	rdgsucopab 3952	. . . . . . . . 9 |- ((y e. On /\ |^|{x e. On | (aleph` y) ~< x} e. V) -> (aleph` suc y) = |^|{x e. On | (aleph` y) ~< x})
20	19	eleq1d 1543	. . . . . . . 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 3019	. . . . . . . 8 |- (|^|{x e. On | (aleph` y) ~< x} e. V <-> |^|{x e. On | (aleph` y) ~< x} e. On)
22	20, 21	syl6rbbr 541	. . . . . . 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 596	. . . . 5 |- (y e. On -> (|^|{x e. On | (aleph` y) ~< x} e. V -> (aleph` suc y) e. On))
25	12, 13, 14, 15, 18	rdgsucopabn 3953	. . . . . 6 |- (-. |^|{x e. On | (aleph` y) ~< x} e. V -> (aleph` suc y) = (/))
26		0elon 3028	. . . . . 6 |- (/) e. On
27	25, 26	syl6eqel 1559	. . . . 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 1816	. . . . . 6 |- x e. V
31		alephlim 4875	. . . . . 6 |- ((x e. V /\ Lim x) -> (aleph` x) = U_y e. x (aleph` y))
32	30, 31	mpan 697	. . . . 5 |- (Lim x -> (aleph` x) = U_y e. x (aleph` y))
33	32	eleq1d 1543	. . . 4 |- (Lim x -> ((aleph` x) e. On <-> U_y e. x (aleph` y) e. On))
34		fvex 3738	. . . . 5 |- (aleph` y) e. V
35	30, 34	iunon 3915	. . . 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 3167	. 2 |- (A e. On -> (aleph` A) e. On)
38		alephfnon 4873	. . . . . . 7 |- aleph Fn On
39		fndm 3593	. . . . . . 7 |- (aleph Fn On -> dom aleph = On)
40	38, 39	ax-mp 7	. . . . . 6 |- dom aleph = On
41	40	eleq2i 1541	. . . . 5 |- (A e. dom aleph <-> A e. On)
42	41	negbii 187	. . . 4 |- (-. A e. dom aleph <-> -. A e. On)
43		ndmfv 3751	. . . 4 |- (-. A e. dom aleph -> (aleph` A) = (/))
44	42, 43	sylbir 201	. . 3 |- (-. A e. On -> (aleph` A) = (/))
45	44, 26	syl6eqel 1559	. 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 958   e. wcel 960  A.wral 1648  {crab 1651  Vcvv 1814  (/)c0 2283  |^|cint 2537  U_ciun 2570   class class class wbr 2624  Oncon0 2954  Lim wlim 2955  suc csuc 2956  omcom 3137  dom cdm 3176   Fn wfn 3183  ` cfv 3188   ~< csdm 4372  alephcale 4824

This theorem is referenced by:  alephnbtwn 4879  alephnbtwn2 4880  alephordlem1 4883  alephordlem2 4884  alephordi 4885  alephord 4886  alephord2 4887  alephord3 4889  alephle 4895  cardaleph 4896  alephfp 4911  alephval2 4913

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-inf2 4634

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-rab 1655  df-v 1815  df-sbc 1945  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  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-fv 3204  df-rdg 3938  df-aleph 4827

Copyright terms: Public domain