Theorem alephnbtwn2 6017

Description: No set has equinumerosity between an aleph and its successor aleph.

Assertion

Ref	Expression
alephnbtwn2	|- -. ((aleph` A) ~< B /\ B ~< (aleph` suc A))

Proof of Theorem alephnbtwn2

Step	Hyp	Ref	Expression
1		cardidm 6001	. . . 4 |- (card` (card` B)) = (card` B)
2		alephnbtwn 6016	. . . 4 |- ((card` (card` B)) = (card` B) -> -. ((aleph` A) e. (card` B) /\ (card` B) e. (aleph` suc A)))
3	1, 2	ax-mp 7	. . 3 |- -. ((aleph` A) e. (card` B) /\ (card` B) e. (aleph` suc A))
4		alephon 5876	. . . . . 6 |- (aleph` A) e. On
5		cardsdomel 6004	. . . . . 6 |- ((aleph` A) e. On -> ((aleph` A) ~< B <-> (aleph` A) e. (card` B)))
6	4, 5	ax-mp 7	. . . . 5 |- ((aleph` A) ~< B <-> (aleph` A) e. (card` B))
7	6	a1i 8	. . . 4 |- (B e. _V -> ((aleph` A) ~< B <-> (aleph` A) e. (card` B)))
8		alephon 5876	. . . . . 6 |- (aleph` suc A) e. On
9		cardsdom 5988	. . . . . 6 |- ((B e. _V /\ (aleph` suc A) e. On) -> ((card` B) e. (card` (aleph` suc A)) <-> B ~< (aleph` suc A)))
10	8, 9	mpan2 760	. . . . 5 |- (B e. _V -> ((card` B) e. (card` (aleph` suc A)) <-> B ~< (aleph` suc A)))
11		alephcard 6015	. . . . . 6 |- (card` (aleph` suc A)) = (aleph` suc A)
12	11	eleq2i 1961	. . . . 5 |- ((card` B) e. (card` (aleph` suc A)) <-> (card` B) e. (aleph` suc A))
13	10, 12	syl5rbbr 594	. . . 4 |- (B e. _V -> (B ~< (aleph` suc A) <-> (card` B) e. (aleph` suc A)))
14	7, 13	anbi12d 690	. . 3 |- (B e. _V -> (((aleph` A) ~< B /\ B ~< (aleph` suc A)) <-> ((aleph` A) e. (card` B) /\ (card` B) e. (aleph` suc A))))
15	3, 14	mtbiri 785	. 2 |- (B e. _V -> -. ((aleph` A) ~< B /\ B ~< (aleph` suc A)))
16		relsdom 5433	. . . . 5 |- Rel ~<
17	16	brrelexi 4029	. . . 4 |- (B ~< (aleph` suc A) -> B e. _V)
18	17	adantl 424	. . 3 |- (((aleph` A) ~< B /\ B ~< (aleph` suc A)) -> B e. _V)
19	18	con3i 114	. 2 |- (-. B e. _V -> -. ((aleph` A) ~< B /\ B ~< (aleph` suc A)))
20	15, 19	pm2.61i 140	1 |- -. ((aleph` A) ~< B /\ B ~< (aleph` suc A))

Syntax hints:  -. wn 2   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  _Vcvv 2292   class class class wbr 3338  Oncon0 3657  suc csuc 3659  ` cfv 3998   ~< csdm 5425  cardccrd 5859  alephcale 5860

This theorem is referenced by:  alephsucpw 6018  alephsucdom 6028  aleph1re 8820

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-reg 5695  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-fin 5430  df-card 5862  df-aleph 5863

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