Theorem alephnbtwn 4840

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 1527	. . . . . . 7 |- ((card` B) = B -> ((aleph` A) e. (card` B) <-> (aleph` A) e. B))
2		alephon 4837	. . . . . . . 8 |- (aleph` A) e. On
3		cardsdomel 4824	. . . . . . . 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 530	. . . . . 6 |- ((card` B) = B -> ((aleph` A) ~< B <-> (aleph` A) e. B))
6	5	adantl 388	. . . . 5 |- ((A e. On /\ (card` B) = B) -> ((aleph` A) ~< B <-> (aleph` A) e. B))
7		alephsuc 4838	. . . . . . . . . 10 |- (A e. On -> (aleph` suc A) = |^|{x e. On | (aleph` A) ~< x})
8	7	eleq2d 1533	. . . . . . . . 9 |- (A e. On -> (B e. (aleph` suc A) <-> B e. |^|{x e. On | (aleph` A) ~< x}))
9	8	biimpd 153	. . . . . . . 8 |- (A e. On -> (B e. (aleph` suc A) -> B e. |^|{x e. On | (aleph` A) ~< x}))
10		breq2 2613	. . . . . . . . 9 |- (x = B -> ((aleph` A) ~< x <-> (aleph` A) ~< B))
11	10	onnminsb 3006	. . . . . . . 8 |- (B e. On -> (B e. |^|{x e. On | (aleph` A) ~< x} -> -. (aleph` A) ~< B))
12	9, 11	sylan9 468	. . . . . . 7 |- ((A e. On /\ B e. On) -> (B e. (aleph` suc A) -> -. (aleph` A) ~< B))
13	12	con2d 91	. . . . . 6 |- ((A e. On /\ B e. On) -> ((aleph` A) ~< B -> -. B e. (aleph` suc A)))
14		cardon 4799	. . . . . . 7 |- (card` B) e. On
15		eleq1 1526	. . . . . . 7 |- ((card` B) = B -> ((card` B) e. On <-> B e. On))
16	14, 15	mpbii 193	. . . . . 6 |- ((card` B) = B -> B e. On)
17	13, 16	sylan2 451	. . . . 5 |- ((A e. On /\ (card` B) = B) -> ((aleph` A) ~< B -> -. B e. (aleph` suc A)))
18	6, 17	sylbird 205	. . . 4 |- ((A e. On /\ (card` B) = B) -> ((aleph` A) e. B -> -. B e. (aleph` suc A)))
19		imnan 242	. . . 4 |- (((aleph` A) e. B -> -. B e. (aleph` suc A)) <-> -. ((aleph` A) e. B /\ B e. (aleph` suc A)))
20	18, 19	sylib 198	. . 3 |- ((A e. On /\ (card` B) = B) -> -. ((aleph` A) e. B /\ B e. (aleph` suc A)))
21	20	ex 373	. 2 |- (A e. On -> ((card` B) = B -> -. ((aleph` A) e. B /\ B e. (aleph` suc A))))
22		n0i 2275	. . . . . . 7 |- (B e. (aleph` suc A) -> -. (aleph` suc A) = (/))
23		alephfnon 4834	. . . . . . . . . . 11 |- aleph Fn On
24		fndm 3573	. . . . . . . . . . 11 |- (aleph Fn On -> dom aleph = On)
25	23, 24	ax-mp 7	. . . . . . . . . 10 |- dom aleph = On
26	25	eleq2i 1530	. . . . . . . . 9 |- (suc A e. dom aleph <-> suc A e. On)
27	26	negbii 187	. . . . . . . 8 |- (-. suc A e. dom aleph <-> -. suc A e. On)
28		ndmfv 3730	. . . . . . . 8 |- (-. suc A e. dom aleph -> (aleph` suc A) = (/))
29	27, 28	sylbir 201	. . . . . . 7 |- (-. suc A e. On -> (aleph` suc A) = (/))
30	22, 29	nsyl2 118	. . . . . 6 |- (B e. (aleph` suc A) -> suc A e. On)
31		sucelon 3058	. . . . . 6 |- (A e. On <-> suc A e. On)
32	30, 31	sylibr 200	. . . . 5 |- (B e. (aleph` suc A) -> A e. On)
33	32	adantl 388	. . . 4 |- (((aleph` A) e. B /\ B e. (aleph` suc A)) -> A e. On)
34	33	con3i 98	. . 3 |- (-. A e. On -> -. ((aleph` A) e. B /\ B e. (aleph` suc A)))
35	34	a1d 12	. 2 |- (-. A e. On -> ((card` B) = B -> -. ((aleph` A) e. B /\ B e. (aleph` suc A))))
36	21, 35	pm2.61i 126	1 |- ((card` B) = B -> -. ((aleph` A) e. B /\ B e. (aleph` suc A)))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   /\ wa 223   = wceq 953   e. wcel 955  {crab 1640  (/)c0 2270  |^|cint 2523   class class class wbr 2609  Oncon0 2938  suc csuc 2940  dom cdm 3160   Fn wfn 3167  ` cfv 3172   ~< csdm 4350  cardccrd 4785  alephcale 4786

This theorem is referenced by:  alephnbtwn2 4841

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  ax-ac 4716

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-reu 1643  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-f 3184  df-f1 3185  df-fo 3186  df-f1o 3187  df-fv 3188  df-rdg 3917  df-er 4245  df-en 4351  df-dom 4352  df-sdom 4353  df-card 4788  df-aleph 4789

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