Theorem alephsucdom 4891

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 4482	. . . . . . 7 |- ((A ~<_ (aleph` B) /\ (aleph` B) ~< (aleph` suc B)) -> A ~< (aleph` suc B))
2	1	ex 373	. . . . . 6 |- (A ~<_ (aleph` B) -> ((aleph` B) ~< (aleph` suc B) -> A ~< (aleph` suc B)))
3		alephordlem1 4883	. . . . . 6 |- (B e. On -> (aleph` B) ~< (aleph` suc B))
4	2, 3	syl5com 52	. . . . 5 |- (B e. On -> (A ~<_ (aleph` B) -> A ~< (aleph` suc B)))
5	4	adantl 390	. . . 4 |- ((A e. V /\ B e. On) -> (A ~<_ (aleph` B) -> A ~< (aleph` suc B)))
6		fvex 3738	. . . . . . 7 |- (aleph` B) e. V
7		domtri 4848	. . . . . . 7 |- ((A e. V /\ (aleph` B) e. V) -> (A ~<_ (aleph` B) <-> -. (aleph` B) ~< A))
8	6, 7	mpan2 698	. . . . . 6 |- (A e. V -> (A ~<_ (aleph` B) <-> -. (aleph` B) ~< A))
9		alephnbtwn2 4880	. . . . . . . 8 |- -. ((aleph` B) ~< A /\ A ~< (aleph` suc B))
10		imnan 242	. . . . . . . 8 |- (((aleph` B) ~< A -> -. A ~< (aleph` suc B)) <-> -. ((aleph` B) ~< A /\ A ~< (aleph` suc B)))
11	9, 10	mpbir 190	. . . . . . 7 |- ((aleph` B) ~< A -> -. A ~< (aleph` suc B))
12	11	con2i 97	. . . . . 6 |- (A ~< (aleph` suc B) -> -. (aleph` B) ~< A)
13	8, 12	syl5bir 210	. . . . 5 |- (A e. V -> (A ~< (aleph` suc B) -> A ~<_ (aleph` B)))
14	13	adantr 391	. . . 4 |- ((A e. V /\ B e. On) -> (A ~< (aleph` suc B) -> A ~<_ (aleph` B)))
15	5, 14	impbid 518	. . 3 |- ((A e. V /\ B e. On) -> (A ~<_ (aleph` B) <-> A ~< (aleph` suc B)))
16	15	ex 373	. 2 |- (A e. V -> (B e. On -> (A ~<_ (aleph` B) <-> A ~< (aleph` suc B))))
17		reldom 4379	. . . . 5 |- Rel ~<_
18	17	brrelexi 3214	. . . 4 |- (A ~<_ (aleph` B) -> A e. V)
19		relsdom 4380	. . . . 5 |- Rel ~<
20	19	brrelexi 3214	. . . 4 |- (A ~< (aleph` suc B) -> A e. V)
21	18, 20	pm5.21ni 680	. . 3 |- (-. A e. V -> (A ~<_ (aleph` B) <-> A ~< (aleph` suc B)))
22	21	a1d 12	. 2 |- (-. A e. V -> (B e. On -> (A ~<_ (aleph` B) <-> A ~< (aleph` suc B))))
23	16, 22	pm2.61i 126	1 |- (B e. On -> (A ~<_ (aleph` B) <-> A ~< (aleph` suc B)))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   /\ wa 223   e. wcel 960  Vcvv 1814   class class class wbr 2624  Oncon0 2954  suc csuc 2956  ` cfv 3188   ~<_ cdom 4371   ~< csdm 4372  alephcale 4824

This theorem is referenced by:  alephsuc2 4892

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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