Theorem alephon 5015

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 3835	. . . 4 |- (x = (/) -> (aleph` x) = (aleph` (/)))
2	1	eleq1d 1583	. . 3 |- (x = (/) -> ((aleph` x) e. On <-> (aleph` (/)) e. On))
3		fveq2 3835	. . . 4 |- (x = y -> (aleph` x) = (aleph` y))
4	3	eleq1d 1583	. . 3 |- (x = y -> ((aleph` x) e. On <-> (aleph` y) e. On))
5		fveq2 3835	. . . 4 |- (x = suc y -> (aleph` x) = (aleph` suc y))
6	5	eleq1d 1583	. . 3 |- (x = suc y -> ((aleph` x) e. On <-> (aleph` suc y) e. On))
7		fveq2 3835	. . . 4 |- (x = A -> (aleph` x) = (aleph` A))
8	7	eleq1d 1583	. . 3 |- (x = A -> ((aleph` x) e. On <-> (aleph` A) e. On))
9		aleph0 5013	. . . 4 |- (aleph` (/)) = om
10		omelon 4775	. . . 4 |- om e. On
11	9, 10	eqeltri 1587	. . 3 |- (aleph` (/)) e. On
12		ax-17 1007	. . . . . . . . . 10 |- (w e. om -> A.z w e. om)
13		ax-17 1007	. . . . . . . . . 10 |- (w e. y -> A.z w e. y)
14		ax-17 1007	. . . . . . . . . 10 |- (w e. |^|{x e. On | (aleph` y) ~< x} -> A.z w e. |^|{x e. On | (aleph` y) ~< x})
15		df-aleph 4963	. . . . . . . . . 10 |- aleph = rec({<.z, y>. | y = |^|{x e. On | z ~< x}}, om)
16		breq1 2695	. . . . . . . . . . . 12 |- (z = (aleph` y) -> (z ~< x <-> (aleph` y) ~< x))
17	16	rabbisdv 1853	. . . . . . . . . . 11 |- (z = (aleph` y) -> {x e. On | z ~< x} = {x e. On | (aleph` y) ~< x})
18	17	inteqd 2605	. . . . . . . . . 10 |- (z = (aleph` y) -> |^|{x e. On | z ~< x} = |^|{x e. On | (aleph` y) ~< x})
19	12, 13, 14, 15, 18	rdgsucopab 4247	. . . . . . . . 9 |- ((y e. On /\ |^|{x e. On | (aleph` y) ~< x} e. V) -> (aleph` suc y) = |^|{x e. On | (aleph` y) ~< x})
20	19	eleq1d 1583	. . . . . . . 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 3158	. . . . . . . 8 |- (|^|{x e. On | (aleph` y) ~< x} e. V <-> |^|{x e. On | (aleph` y) ~< x} e. On)
22	20, 21	syl6rbbr 542	. . . . . . 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 371	. . . . . 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 597	. . . . 5 |- (y e. On -> (|^|{x e. On | (aleph` y) ~< x} e. V -> (aleph` suc y) e. On))
25	12, 13, 14, 15, 18	rdgsucopabn 4248	. . . . . 6 |- (-. |^|{x e. On | (aleph` y) ~< x} e. V -> (aleph` suc y) = (/))
26		0elon 3026	. . . . . 6 |- (/) e. On
27	25, 26	syl6eqel 1599	. . . . 5 |- (-. |^|{x e. On | (aleph` y) ~< x} e. V -> (aleph` suc y) e. On)
28	24, 27	pm2.61d1 126	. . . 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 1859	. . . . . 6 |- x e. V
31		alephlim 5014	. . . . . 6 |- ((x e. V /\ Lim x) -> (aleph` x) = U_y e. x (aleph` y))
32	30, 31	mpan 699	. . . . 5 |- (Lim x -> (aleph` x) = U_y e. x (aleph` y))
33	32	eleq1d 1583	. . . 4 |- (Lim x -> ((aleph` x) e. On <-> U_y e. x (aleph` y) e. On))
34		fvex 3843	. . . . 5 |- (aleph` y) e. V
35	30, 34	iunon 4207	. . . 4 |- (A.y e. x (aleph` y) e. On -> U_y e. x (aleph` y) e. On)
36	33, 35	syl5bir 208	. . 3 |- (Lim x -> (A.y e. x (aleph` y) e. On -> (aleph` x) e. On))
37	2, 4, 6, 8, 11, 29, 36	tfinds 3212	. 2 |- (A e. On -> (aleph` A) e. On)
38		alephfnon 5012	. . . . . . 7 |- aleph Fn On
39		fndm 3693	. . . . . . 7 |- (aleph Fn On -> dom aleph = On)
40	38, 39	ax-mp 7	. . . . . 6 |- dom aleph = On
41	40	eleq2i 1581	. . . . 5 |- (A e. dom aleph <-> A e. On)
42	41	notbii 185	. . . 4 |- (-. A e. dom aleph <-> -. A e. On)
43		ndmfv 3856	. . . 4 |- (-. A e. dom aleph -> (aleph` A) = (/))
44	42, 43	sylbir 199	. . 3 |- (-. A e. On -> (aleph` A) = (/))
45	44, 26	syl6eqel 1599	. 2 |- (-. A e. On -> (aleph` A) e. On)
46	37, 45	pm2.61i 124	1 |- (aleph` A) e. On

Syntax hints:  -. wn 2   <-> wb 144   /\ wa 221   = wceq 992   e. wcel 994  A.wral 1691  {crab 1694  Vcvv 1857  (/)c0 2332  |^|cint 2600  U_ciun 2633   class class class wbr 2692  Oncon0 2975  Lim wlim 2976  suc csuc 2977  omcom 3218  dom cdm 3251   Fn wfn 3258  ` cfv 3263   ~< csdm 4507  alephcale 4960

This theorem is referenced by:  alephnbtwn 5018  alephnbtwn2 5019  alephordlem1 5022  alephordlem2 5023  alephordi 5024  alephord 5025  alephord2 5026  alephord3 5028  alephle 5034  cardaleph 5035  alephfp 5050  alephval2 5052  omsubsuc 11438  omsubsuc2 11439  omsubsdomlem1 11440  omsubel 11444  omsubss 11445  elomsubsd 11446  omsubdmss 11447  omsublim 11448  omsubindss 11449  infenomsub 11450  omsubinit 11451

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 998  ax-gen 999  ax-8 1000  ax-9 1001  ax-10 1002  ax-11 1003  ax-12 1004  ax-13 1005  ax-14 1006  ax-17 1007  ax-4 1009  ax-5o 1011  ax-6o 1014  ax-9o 1159  ax-10o 1177  ax-16 1247  ax-11o 1255  ax-ext 1500  ax-rep 2767  ax-sep 2777  ax-nul 2784  ax-pow 2818  ax-pr 2855  ax-un 3089  ax-inf2 4770

This theorem depends on definitions:  df-bi 145  df-or 222  df-an 223  df-3or 782  df-3an 783  df-ex 1017  df-sb 1209  df-eu 1421  df-mo 1422  df-clab 1506  df-cleq 1511  df-clel 1514  df-ne 1630  df-ral 1695  df-rex 1696  df-rab 1698  df-v 1858  df-sbc 1987  df-csb 2052  df-dif 2101  df-un 2102  df-in 2103  df-ss 2105  df-nul 2333  df-if 2416  df-pw 2459  df-sn 2470  df-pr 2471  df-tp 2473  df-op 2474  df-uni 2570  df-int 2601  df-iun 2635  df-br 2693  df-opab 2741  df-tr 2755  df-eprel 2910  df-id 2913  df-po 2918  df-so 2929  df-fr 2947  df-we 2962  df-ord 2978  df-on 2979  df-lim 2980  df-suc 2981  df-om 3219  df-xp 3265  df-rel 3266  df-cnv 3267  df-co 3268  df-dm 3269  df-rn 3270  df-res 3271  df-ima 3272  df-fun 3273  df-fn 3274  df-fv 3279  df-rdg 4233  df-aleph 4963

Copyright terms: Public domain