Theorem alephnbtwn2 4869

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 4849	. . . 4 |- (card` (card` B)) = (card` B)
2		alephnbtwn 4868	. . . 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 4865	. . . . . 6 |- (aleph` A) e. On
5		cardsdomel 4852	. . . . . 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 4865	. . . . . 6 |- (aleph` suc A) e. On
9		cardsdom 4837	. . . . . 6 |- ((B e. V /\ (aleph` suc A) e. On) -> ((card` B) e. (card` (aleph` suc A)) <-> B ~< (aleph` suc A)))
10	8, 9	mpan2 696	. . . . 5 |- (B e. V -> ((card` B) e. (card` (aleph` suc A)) <-> B ~< (aleph` suc A)))
11		alephcard 4867	. . . . . 6 |- (card` (aleph` suc A)) = (aleph` suc A)
12	11	eleq2i 1538	. . . . 5 |- ((card` B) e. (card` (aleph` suc A)) <-> (card` B) e. (aleph` suc A))
13	10, 12	syl5rbbr 535	. . . 4 |- (B e. V -> (B ~< (aleph` suc A) <-> (card` B) e. (aleph` suc A)))
14	7, 13	anbi12d 628	. . 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 717	. 2 |- (B e. V -> -. ((aleph` A) ~< B /\ B ~< (aleph` suc A)))
16		relsdom 4374	. . . . 5 |- Rel ~<
17	16	brrelexi 3208	. . . 4 |- (B ~< (aleph` suc A) -> B e. V)
18	17	adantl 388	. . 3 |- (((aleph` A) ~< B /\ B ~< (aleph` suc A)) -> B e. V)
19	18	con3i 98	. 2 |- (-. B e. V -> -. ((aleph` A) ~< B /\ B ~< (aleph` suc A)))
20	15, 19	pm2.61i 126	1 |- -. ((aleph` A) ~< B /\ B ~< (aleph` suc A))

Syntax hints:  -. wn 2   <-> wb 146   /\ wa 223   = wceq 956   e. wcel 958  Vcvv 1811   class class class wbr 2619  Oncon0 2948  suc csuc 2950  ` cfv 3182   ~< csdm 4366  cardccrd 4813  alephcale 4814

This theorem is referenced by:  alephsucpw 4870  alephsucdom 4880  aleph1re 7551

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-reg 4593  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-pss 2055  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

Copyright terms: Public domain