Theorem alephnbtwn 6016

Description: No cardinal can be sandwiched between an aleph and its successor aleph. Theorem 67 of [Suppes] p. 229.

Assertion

Ref	Expression
alephnbtwn	|- ((card` B) = B -> -. ((aleph` A) e. B /\ B e. (aleph` suc A)))

Proof of Theorem alephnbtwn

Step	Hyp	Ref	Expression
1		eleq2 1958	. . . . . . 7 |- ((card` B) = B -> ((aleph` A) e. (card` B) <-> (aleph` A) e. B))
2		alephon 5876	. . . . . . . 8 |- (aleph` A) e. On
3		cardsdomel 6004	. . . . . . . 8 |- ((aleph` A) e. On -> ((aleph` A) ~< B <-> (aleph` A) e. (card` B)))
4	2, 3	ax-mp 7	. . . . . . 7 |- ((aleph` A) ~< B <-> (aleph` A) e. (card` B))
5	1, 4	syl5bb 591	. . . . . 6 |- ((card` B) = B -> ((aleph` A) ~< B <-> (aleph` A) e. B))
6	5	adantl 424	. . . . 5 |- ((A e. On /\ (card` B) = B) -> ((aleph` A) ~< B <-> (aleph` A) e. B))
7		alephsuc 6014	. . . . . . . . . 10 |- (A e. On -> (aleph` suc A) = |^|{x e. On | (aleph` A) ~< x})
8	7	eleq2d 1964	. . . . . . . . 9 |- (A e. On -> (B e. (aleph` suc A) <-> B e. |^|{x e. On | (aleph` A) ~< x}))
9	8	biimpd 170	. . . . . . . 8 |- (A e. On -> (B e. (aleph` suc A) -> B e. |^|{x e. On | (aleph` A) ~< x}))
10		breq2 3342	. . . . . . . . 9 |- (x = B -> ((aleph` A) ~< x <-> (aleph` A) ~< B))
11	10	onnminsb 3885	. . . . . . . 8 |- (B e. On -> (B e. |^|{x e. On | (aleph` A) ~< x} -> -. (aleph` A) ~< B))
12	9, 11	sylan9 517	. . . . . . 7 |- ((A e. On /\ B e. On) -> (B e. (aleph` suc A) -> -. (aleph` A) ~< B))
13	12	con2d 107	. . . . . 6 |- ((A e. On /\ B e. On) -> ((aleph` A) ~< B -> -. B e. (aleph` suc A)))
14		cardon 5976	. . . . . . 7 |- (card` B) e. On
15		eleq1 1957	. . . . . . 7 |- ((card` B) = B -> ((card` B) e. On <-> B e. On))
16	14, 15	mpbii 210	. . . . . 6 |- ((card` B) = B -> B e. On)
17	13, 16	sylan2 500	. . . . 5 |- ((A e. On /\ (card` B) = B) -> ((aleph` A) ~< B -> -. B e. (aleph` suc A)))
18	6, 17	sylbird 222	. . . 4 |- ((A e. On /\ (card` B) = B) -> ((aleph` A) e. B -> -. B e. (aleph` suc A)))
19		imnan 261	. . . 4 |- (((aleph` A) e. B -> -. B e. (aleph` suc A)) <-> -. ((aleph` A) e. B /\ B e. (aleph` suc A)))
20	18, 19	sylib 215	. . 3 |- ((A e. On /\ (card` B) = B) -> -. ((aleph` A) e. B /\ B e. (aleph` suc A)))
21	20	ex 402	. 2 |- (A e. On -> ((card` B) = B -> -. ((aleph` A) e. B /\ B e. (aleph` suc A))))
22		n0i 2880	. . . . . . 7 |- (B e. (aleph` suc A) -> -. (aleph` suc A) = (/))
23		alephfnon 5873	. . . . . . . . . . 11 |- aleph Fn On
24		fndm 4512	. . . . . . . . . . 11 |- (aleph Fn On -> dom aleph = On)
25	23, 24	ax-mp 7	. . . . . . . . . 10 |- dom aleph = On
26	25	eleq2i 1961	. . . . . . . . 9 |- (suc A e. dom aleph <-> suc A e. On)
27	26	notbii 204	. . . . . . . 8 |- (-. suc A e. dom aleph <-> -. suc A e. On)
28		ndmfv 4702	. . . . . . . 8 |- (-. suc A e. dom aleph -> (aleph` suc A) = (/))
29	27, 28	sylbir 218	. . . . . . 7 |- (-. suc A e. On -> (aleph` suc A) = (/))
30	22, 29	nsyl2 133	. . . . . 6 |- (B e. (aleph` suc A) -> suc A e. On)
31		sucelon 3898	. . . . . 6 |- (A e. On <-> suc A e. On)
32	30, 31	sylibr 217	. . . . 5 |- (B e. (aleph` suc A) -> A e. On)
33	32	adantl 424	. . . 4 |- (((aleph` A) e. B /\ B e. (aleph` suc A)) -> A e. On)
34	33	con3i 114	. . 3 |- (-. A e. On -> -. ((aleph` A) e. B /\ B e. (aleph` suc A)))
35	34	a1d 15	. 2 |- (-. A e. On -> ((card` B) = B -> -. ((aleph` A) e. B /\ B e. (aleph` suc A))))
36	21, 35	pm2.61i 140	1 |- ((card` B) = B -> -. ((aleph` A) e. B /\ B e. (aleph` suc A)))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  {crab 2108  (/)c0 2875  |^|cint 3214   class class class wbr 3338  Oncon0 3657  suc csuc 3659  dom cdm 3986   Fn wfn 3993  ` cfv 3998   ~< csdm 5425  cardccrd 5859  alephcale 5860

This theorem is referenced by:  alephnbtwn2 6017

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  ax-ac 5906

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-reu 2111  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-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-rdg 5140  df-er 5318  df-en 5427  df-dom 5428  df-sdom 5429  df-card 5862  df-aleph 5863

Copyright terms: Public domain