Theorem alephsucdom 6028

Description: A set dominated by an aleph is strictly dominated by its successor aleph and vice-versa.

Assertion

Ref	Expression
alephsucdom	|- (B e. On -> (A ~<_ (aleph` B) <-> A ~< (aleph` suc B)))

Proof of Theorem alephsucdom

Step	Hyp	Ref	Expression
1		domsdomtr 5539	. . . . . . 7 |- ((A ~<_ (aleph` B) /\ (aleph` B) ~< (aleph` suc B)) -> A ~< (aleph` suc B))
2	1	ex 402	. . . . . 6 |- (A ~<_ (aleph` B) -> ((aleph` B) ~< (aleph` suc B) -> A ~< (aleph` suc B)))
3		alephordlem1 6020	. . . . . 6 |- (B e. On -> (aleph` B) ~< (aleph` suc B))
4	2, 3	syl5com 63	. . . . 5 |- (B e. On -> (A ~<_ (aleph` B) -> A ~< (aleph` suc B)))
5	4	adantl 424	. . . 4 |- ((A e. _V /\ B e. On) -> (A ~<_ (aleph` B) -> A ~< (aleph` suc B)))
6		fvex 4689	. . . . . . 7 |- (aleph` B) e. _V
7		domtri 5989	. . . . . . 7 |- ((A e. _V /\ (aleph` B) e. _V) -> (A ~<_ (aleph` B) <-> -. (aleph` B) ~< A))
8	6, 7	mpan2 760	. . . . . 6 |- (A e. _V -> (A ~<_ (aleph` B) <-> -. (aleph` B) ~< A))
9		alephnbtwn2 6017	. . . . . . . 8 |- -. ((aleph` B) ~< A /\ A ~< (aleph` suc B))
10		imnan 261	. . . . . . . 8 |- (((aleph` B) ~< A -> -. A ~< (aleph` suc B)) <-> -. ((aleph` B) ~< A /\ A ~< (aleph` suc B)))
11	9, 10	mpbir 207	. . . . . . 7 |- ((aleph` B) ~< A -> -. A ~< (aleph` suc B))
12	11	con2i 113	. . . . . 6 |- (A ~< (aleph` suc B) -> -. (aleph` B) ~< A)
13	8, 12	syl5bir 227	. . . . 5 |- (A e. _V -> (A ~< (aleph` suc B) -> A ~<_ (aleph` B)))
14	13	adantr 425	. . . 4 |- ((A e. _V /\ B e. On) -> (A ~< (aleph` suc B) -> A ~<_ (aleph` B)))
15	5, 14	impbid 574	. . 3 |- ((A e. _V /\ B e. On) -> (A ~<_ (aleph` B) <-> A ~< (aleph` suc B)))
16	15	ex 402	. 2 |- (A e. _V -> (B e. On -> (A ~<_ (aleph` B) <-> A ~< (aleph` suc B))))
17		reldom 5432	. . . . 5 |- Rel ~<_
18	17	brrelexi 4029	. . . 4 |- (A ~<_ (aleph` B) -> A e. _V)
19		relsdom 5433	. . . . 5 |- Rel ~<
20	19	brrelexi 4029	. . . 4 |- (A ~< (aleph` suc B) -> A e. _V)
21	18, 20	pm5.21ni 742	. . 3 |- (-. A e. _V -> (A ~<_ (aleph` B) <-> A ~< (aleph` suc B)))
22	21	a1d 15	. 2 |- (-. A e. _V -> (B e. On -> (A ~<_ (aleph` B) <-> A ~< (aleph` suc B))))
23	16, 22	pm2.61i 140	1 |- (B e. On -> (A ~<_ (aleph` B) <-> A ~< (aleph` suc B)))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 163   /\ wa 240   e. wcel 1300  _Vcvv 2292   class class class wbr 3338  Oncon0 3657  suc csuc 3659  ` cfv 3998   ~<_ cdom 5424   ~< csdm 5425  alephcale 5860

This theorem is referenced by:  alephsuc2 6029

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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