Theorem alephsucdom 4852

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 4456	. . . . . . 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 4844	. . . . . 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 388	. . . 4 |- ((A e. V /\ B e. On) -> (A ~<_ (aleph` B) -> A ~< (aleph` suc B)))
6		fvex 3717	. . . . . . 7 |- (aleph` B) e. V
7		domtri 4810	. . . . . . 7 |- ((A e. V /\ (aleph` B) e. V) -> (A ~<_ (aleph` B) <-> -. (aleph` B) ~< A))
8	6, 7	mpan2 694	. . . . . 6 |- (A e. V -> (A ~<_ (aleph` B) <-> -. (aleph` B) ~< A))
9		alephnbtwn2 4841	. . . . . . . 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 389	. . . 4 |- ((A e. V /\ B e. On) -> (A ~< (aleph` suc B) -> A ~<_ (aleph` B)))
15	5, 14	impbid 514	. . 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 4355	. . . . 5 |- Rel ~<_
18	17	brrelexi 3198	. . . 4 |- (A ~<_ (aleph` B) -> A e. V)
19		relsdom 4356	. . . . 5 |- Rel ~<
20	19	brrelexi 3198	. . . 4 |- (A ~< (aleph` suc B) -> A e. V)
21	18, 20	pm5.21ni 676	. . 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 955  Vcvv 1802   class class class wbr 2609  Oncon0 2938  suc csuc 2940  ` cfv 3172   ~<_ cdom 4349   ~< csdm 4350  alephcale 4786

This theorem is referenced by:  alephsuc2 4853

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-rep 2683  ax-sep 2693  ax-nul 2700  ax-pow 2732  ax-pr 2769  ax-un 2857  ax-reg 4565  ax-inf2 4597  ax-ac 4716

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 774  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-reu 1643  df-rab 1644  df-v 1803  df-sbc 1932  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-pss 2045  df-nul 2271  df-if 2352  df-pw 2392  df-sn 2402  df-pr 2403  df-tp 2405  df-op 2406  df-uni 2494  df-int 2524  df-iun 2558  df-br 2610  df-opab 2657  df-tr 2671  df-eprel 2821  df-id 2824  df-po 2831  df-so 2841  df-fr 2907  df-we 2924  df-ord 2941  df-on 2942  df-lim 2943  df-suc 2944  df-om 3122  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-f 3184  df-f1 3185  df-fo 3186  df-f1o 3187  df-fv 3188  df-rdg 3917  df-er 4245  df-en 4351  df-dom 4352  df-sdom 4353  df-card 4788  df-aleph 4789

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