Theorem alephsucdom 3685

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 3374	. . . . . . . 8 |- ((A ~<_ (aleph` B) /\ (aleph` B) ~< (aleph` suc B)) -> A ~< (aleph` suc B))
2	1	exp 291	. . . . . . 7 |- (A ~<_ (aleph` B) -> ((aleph` B) ~< (aleph` suc B) -> A ~< (aleph` suc B)))
3		alephordlem1 3677	. . . . . . 7 |- (B e. On -> (aleph` B) ~< (aleph` suc B))
4	2, 3	syl5 22	. . . . . 6 |- (A ~<_ (aleph` B) -> (B e. On -> A ~< (aleph` suc B)))
5	4	com12 13	. . . . 5 |- (B e. On -> (A ~<_ (aleph` B) -> A ~< (aleph` suc B)))
6	5	adantl 305	. . . 4 |- ((A e. V /\ B e. On) -> (A ~<_ (aleph` B) -> A ~< (aleph` suc B)))
7		fvex 2838	. . . . . . 7 |- (aleph` B) e. V
8		domtri 3644	. . . . . . 7 |- ((A e. V /\ (aleph` B) e. V) -> (A ~<_ (aleph` B) <-> -. (aleph` B) ~< A))
9	7, 8	mpan2 519	. . . . . 6 |- (A e. V -> (A ~<_ (aleph` B) <-> -. (aleph` B) ~< A))
10		alephnbtwn2 3675	. . . . . . . 8 |- -. ((aleph` B) ~< A /\ A ~< (aleph` suc B))
11		imnan 207	. . . . . . . 8 |- (((aleph` B) ~< A -> -. A ~< (aleph` suc B)) <-> -. ((aleph` B) ~< A /\ A ~< (aleph` suc B)))
12	10, 11	mpbir 165	. . . . . . 7 |- ((aleph` B) ~< A -> -. A ~< (aleph` suc B))
13	12	con2i 89	. . . . . 6 |- (A ~< (aleph` suc B) -> -. (aleph` B) ~< A)
14	9, 13	syl5bir 184	. . . . 5 |- (A e. V -> (A ~< (aleph` suc B) -> A ~<_ (aleph` B)))
15	14	adantr 306	. . . 4 |- ((A e. V /\ B e. On) -> (A ~< (aleph` suc B) -> A ~<_ (aleph` B)))
16	6, 15	impbid 397	. . 3 |- ((A e. V /\ B e. On) -> (A ~<_ (aleph` B) <-> A ~< (aleph` suc B)))
17	16	exp 291	. 2 |- (A e. V -> (B e. On -> (A ~<_ (aleph` B) <-> A ~< (aleph` suc B))))
18		reldom 3278	. . . . 5 |- Rel ~<_
19	18	brrelexi 2447	. . . 4 |- (A ~<_ (aleph` B) -> A e. V)
20		relsdom 3279	. . . . 5 |- Rel ~<
21	20	brrelexi 2447	. . . 4 |- (A ~< (aleph` suc B) -> A e. V)
22	19, 21	pm5.21ni 503	. . 3 |- (-. A e. V -> (A ~<_ (aleph` B) <-> A ~< (aleph` suc B)))
23	22	a1d 14	. 2 |- (-. A e. V -> (B e. On -> (A ~<_ (aleph` B) <-> A ~< (aleph` suc B))))
24	17, 23	pm2.61i 110	1 |- (B e. On -> (A ~<_ (aleph` B) <-> A ~< (aleph` suc B)))

Syntax hints:  -. wn 1   -> wi 2   <-> wb 127   /\ wa 196   e. wcel 1092  Vcvv 1348   class class class wbr 2054  Oncon0 2199  suc csuc 2201  ` cfv 2422   ~<_ cdom 3272   ~< csdm 3273  alephcale 3621

This theorem is referenced by:  alephsuc2 3686

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077  ax-reg 1078  ax-inf 1079  ax-ac 1080

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ne 1192  df-ral 1205  df-rex 1206  df-reu 1207  df-rab 1208  df-v 1349  df-sbc 1441  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-pss 1494  df-nul 1708  df-if 1777  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-int 1966  df-iun 1996  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-lim 2204  df-suc 2205  df-om 2373  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fo 2436  df-f1o 2437  df-fv 2438  df-rdg 2970  df-er 3200  df-en 3274  df-dom 3275  df-sdom 3276  df-card 3623  df-aleph 3624

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