Theorem aleph1 3676

Description: The set exponentiation of 2 to the aleph-zero has cardinality of at least aleph-one. (If we were to assume the Continuum Hypothesis, their cardinalities would be the same.)

Assertion

Ref	Expression
aleph1	|- (aleph` 1o) ~<_ (2o ^m (aleph` (/)))

Proof of Theorem aleph1

Step	Hyp	Ref	Expression
1		df-1o 3104	. . 3 |- 1o = suc (/)
2	1	fveq2i 2835	. 2 |- (aleph` 1o) = (aleph` suc (/))
3		fvex 2838	. . . . . 6 |- (aleph` (/)) e. V
4	3	canth2 3381	. . . . 5 |- (aleph` (/)) ~< P~(aleph` (/))
5		alephnbtwn2 3675	. . . . . 6 |- -. ((aleph` (/)) ~< P~(aleph` (/)) /\ P~(aleph` (/)) ~< (aleph` suc (/)))
6		imnan 207	. . . . . 6 |- (((aleph` (/)) ~< P~(aleph` (/)) -> -. P~(aleph` (/)) ~< (aleph` suc (/))) <-> -. ((aleph` (/)) ~< P~(aleph` (/)) /\ P~(aleph` (/)) ~< (aleph` suc (/))))
7	5, 6	mpbir 165	. . . . 5 |- ((aleph` (/)) ~< P~(aleph` (/)) -> -. P~(aleph` (/)) ~< (aleph` suc (/)))
8	4, 7	ax-mp 6	. . . 4 |- -. P~(aleph` (/)) ~< (aleph` suc (/))
9		fvex 2838	. . . . 5 |- (aleph` suc (/)) e. V
10	3	pwex 1806	. . . . 5 |- P~(aleph` (/)) e. V
11		domtri 3644	. . . . 5 |- (((aleph` suc (/)) e. V /\ P~(aleph` (/)) e. V) -> ((aleph` suc (/)) ~<_ P~(aleph` (/)) <-> -. P~(aleph` (/)) ~< (aleph` suc (/))))
12	9, 10, 11	mp2an 520	. . . 4 |- ((aleph` suc (/)) ~<_ P~(aleph` (/)) <-> -. P~(aleph` (/)) ~< (aleph` suc (/)))
13	8, 12	mpbir 165	. . 3 |- (aleph` suc (/)) ~<_ P~(aleph` (/))
14		oprex 3018	. . . 4 |- (2o ^m (aleph` (/))) e. V
15	3	pw2en 3348	. . . 4 |- P~(aleph` (/)) ~~ (2o ^m (aleph` (/)))
16		domen2 3378	. . . 4 |- (((2o ^m (aleph` (/))) e. V /\ P~(aleph` (/)) ~~ (2o ^m (aleph` (/)))) -> ((aleph` suc (/)) ~<_ P~(aleph` (/)) <-> (aleph` suc (/)) ~<_ (2o ^m (aleph` (/)))))
17	14, 15, 16	mp2an 520	. . 3 |- ((aleph` suc (/)) ~<_ P~(aleph` (/)) <-> (aleph` suc (/)) ~<_ (2o ^m (aleph` (/))))
18	13, 17	mpbi 164	. 2 |- (aleph` suc (/)) ~<_ (2o ^m (aleph` (/)))
19	2, 18	eqbrtr 2076	1 |- (aleph` 1o) ~<_ (2o ^m (aleph` (/)))

Syntax hints:  -. wn 1   -> wi 2   <-> wb 127   /\ wa 196   e. wcel 1092  Vcvv 1348  (/)c0 1707  P~cpw 1798   class class class wbr 2054  suc csuc 2201  ` cfv 2422  (class class class)co 3001  1oc1o 3099  2oc2o 3100   ^m cm 3258   ~~ cen 3271   ~<_ cdom 3272   ~< csdm 3273  alephcale 3621

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-opr 3003  df-oprab 3004  df-1o 3104  df-2o 3105  df-er 3200  df-map 3259  df-en 3274  df-dom 3275  df-sdom 3276  df-card 3623  df-aleph 3624

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