Theorem alephnbtwn2 4880

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 4860	. . . 4 |- (card` (card` B)) = (card` B)
2		alephnbtwn 4879	. . . 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 4876	. . . . . 6 |- (aleph` A) e. On
5		cardsdomel 4863	. . . . . 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 4876	. . . . . 6 |- (aleph` suc A) e. On
9		cardsdom 4847	. . . . . 6 |- ((B e. V /\ (aleph` suc A) e. On) -> ((card` B) e. (card` (aleph` suc A)) <-> B ~< (aleph` suc A)))
10	8, 9	mpan2 698	. . . . 5 |- (B e. V -> ((card` B) e. (card` (aleph` suc A)) <-> B ~< (aleph` suc A)))
11		alephcard 4878	. . . . . 6 |- (card` (aleph` suc A)) = (aleph` suc A)
12	11	eleq2i 1541	. . . . 5 |- ((card` B) e. (card` (aleph` suc A)) <-> (card` B) e. (aleph` suc A))
13	10, 12	syl5rbbr 537	. . . 4 |- (B e. V -> (B ~< (aleph` suc A) <-> (card` B) e. (aleph` suc A)))
14	7, 13	anbi12d 630	. . 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 719	. 2 |- (B e. V -> -. ((aleph` A) ~< B /\ B ~< (aleph` suc A)))
16		relsdom 4380	. . . . 5 |- Rel ~<
17	16	brrelexi 3214	. . . 4 |- (B ~< (aleph` suc A) -> B e. V)
18	17	adantl 390	. . 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 958   e. wcel 960  Vcvv 1814   class class class wbr 2624  Oncon0 2954  suc csuc 2956  ` cfv 3188   ~< csdm 4372  cardccrd 4823  alephcale 4824

This theorem is referenced by:  alephsucpw 4881  alephsucdom 4891  aleph1re 7552

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-rep 2698  ax-sep 2708  ax-nul 2715  ax-pow 2748  ax-pr 2785  ax-un 2872  ax-reg 4602  ax-inf2 4634  ax-ac 4754

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 778  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-reu 1654  df-rab 1655  df-v 1815  df-sbc 1945  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-pss 2058  df-nul 2284  df-if 2366  df-pw 2406  df-sn 2416  df-pr 2417  df-tp 2419  df-op 2420  df-uni 2508  df-int 2538  df-iun 2572  df-br 2625  df-opab 2672  df-tr 2686  df-eprel 2838  df-id 2841  df-po 2846  df-so 2856  df-fr 2923  df-we 2940  df-ord 2957  df-on 2958  df-lim 2959  df-suc 2960  df-om 3138  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-f 3200  df-f1 3201  df-fo 3202  df-f1o 3203  df-fv 3204  df-rdg 3938  df-er 4267  df-en 4374  df-dom 4375  df-sdom 4376  df-fin 4377  df-card 4826  df-aleph 4827

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