Theorem alephnbtwn 4868

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 1535	. . . . . . 7 |- ((card` B) = B -> ((aleph` A) e. (card` B) <-> (aleph` A) e. B))
2		alephon 4865	. . . . . . . 8 |- (aleph` A) e. On
3		cardsdomel 4852	. . . . . . . 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 532	. . . . . 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 4866	. . . . . . . . . 10 |- (A e. On -> (aleph` suc A) = |^|{x e. On | (aleph` A) ~< x})
8	7	eleq2d 1541	. . . . . . . . 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 2623	. . . . . . . . 9 |- (x = B -> ((aleph` A) ~< x <-> (aleph` A) ~< B))
11	10	onnminsb 3016	. . . . . . . 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 4827	. . . . . . 7 |- (card` B) e. On
15		eleq1 1534	. . . . . . 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 2285	. . . . . . 7 |- (B e. (aleph` suc A) -> -. (aleph` suc A) = (/))
23		alephfnon 4862	. . . . . . . . . . 11 |- aleph Fn On
24		fndm 3587	. . . . . . . . . . 11 |- (aleph Fn On -> dom aleph = On)
25	23, 24	ax-mp 7	. . . . . . . . . 10 |- dom aleph = On
26	25	eleq2i 1538	. . . . . . . . 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 3745	. . . . . . . 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 3068	. . . . . 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 956   e. wcel 958  {crab 1648  (/)c0 2280  |^|cint 2533   class class class wbr 2619  Oncon0 2948  suc csuc 2950  dom cdm 3170   Fn wfn 3177  ` cfv 3182   ~< csdm 4366  cardccrd 4813  alephcale 4814

This theorem is referenced by:  alephnbtwn2 4869

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2693  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866  ax-inf2 4625  ax-ac 4744

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 776  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-reu 1651  df-rab 1652  df-v 1812  df-sbc 1942  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-if 2362  df-pw 2402  df-sn 2412  df-pr 2413  df-tp 2415  df-op 2416  df-uni 2504  df-int 2534  df-iun 2568  df-br 2620  df-opab 2667  df-tr 2681  df-eprel 2832  df-id 2835  df-po 2840  df-so 2850  df-fr 2917  df-we 2934  df-ord 2951  df-on 2952  df-lim 2953  df-suc 2954  df-om 3132  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-f 3194  df-f1 3195  df-fo 3196  df-f1o 3197  df-fv 3198  df-rdg 3932  df-er 4261  df-en 4368  df-dom 4369  df-sdom 4370  df-card 4816  df-aleph 4817

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