Theorem alephnbtwn2 4792

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 4772	. . . 4 |- (card` (card` B)) = (card` B)
2		alephnbtwn 4791	. . . 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 4788	. . . . . 6 |- (aleph` A) e. On
5		cardsdomel 4775	. . . . . 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 4788	. . . . . 6 |- (aleph` suc A) e. On
9		cardsdom 4760	. . . . . 6 |- ((B e. V /\ (aleph` suc A) e. On) -> ((card` B) e. (card` (aleph` suc A)) <-> B ~< (aleph` suc A)))
10	8, 9	mpan2 693	. . . . 5 |- (B e. V -> ((card` B) e. (card` (aleph` suc A)) <-> B ~< (aleph` suc A)))
11		alephcard 4790	. . . . . 6 |- (card` (aleph` suc A)) = (aleph` suc A)
12	11	eleq2i 1514	. . . . 5 |- ((card` B) e. (card` (aleph` suc A)) <-> (card` B) e. (aleph` suc A))
13	10, 12	syl5rbbr 533	. . . 4 |- (B e. V -> (B ~< (aleph` suc A) <-> (card` B) e. (aleph` suc A)))
14	7, 13	anbi12d 626	. . 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 714	. 2 |- (B e. V -> -. ((aleph` A) ~< B /\ B ~< (aleph` suc A)))
16		relsdom 4310	. . . . 5 |- Rel ~<
17	16	brrelexi 3170	. . . 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 1099   e. wcel 1105  Vcvv 1786   class class class wbr 2587  Oncon0 2911  suc csuc 2913  ` cfv 3145   ~< csdm 4304  cardccrd 4737  alephcale 4738

This theorem is referenced by:  alephsucpw 4793  alephsucdom 4803  aleph1re 7445

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-4 951  ax-5 952  ax-6 953  ax-7 954  ax-gen 955  ax-8 1101  ax-9 1102  ax-10 1103  ax-12 1104  ax-13 1107  ax-14 1108  ax-11 1180  ax-17 1190  ax-16 1194  ax-11o 1202  ax-ext 1436  ax-rep 2661  ax-sep 2671  ax-nul 2678  ax-pow 2710  ax-pr 2747  ax-un 2830  ax-reg 4517  ax-inf2 4549  ax-ac 4668

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 773  df-3an 774  df-ex 957  df-sb 1155  df-eu 1359  df-mo 1360  df-clab 1441  df-cleq 1446  df-clel 1449  df-ne 1563  df-ral 1625  df-rex 1626  df-reu 1627  df-rab 1628  df-v 1787  df-sbc 1913  df-dif 2020  df-un 2021  df-in 2022  df-ss 2024  df-pss 2026  df-nul 2252  df-if 2333  df-pw 2373  df-sn 2383  df-pr 2384  df-tp 2386  df-op 2387  df-uni 2472  df-int 2502  df-iun 2536  df-br 2588  df-opab 2635  df-tr 2649  df-eprel 2794  df-id 2797  df-po 2804  df-so 2814  df-fr 2880  df-we 2897  df-ord 2914  df-on 2915  df-lim 2916  df-suc 2917  df-om 3095  df-xp 3147  df-rel 3148  df-cnv 3149  df-co 3150  df-dm 3151  df-rn 3152  df-res 3153  df-ima 3154  df-fun 3155  df-fn 3156  df-f 3157  df-f1 3158  df-fo 3159  df-f1o 3160  df-fv 3161  df-rdg 3871  df-er 4199  df-en 4305  df-dom 4306  df-sdom 4307  df-card 4740  df-aleph 4741

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