Theorem alephsuc 3672

Description: Value of the aleph function at a successor ordinal. Definition 12(ii) of [Suppes] p. 91.

Assertion

Ref	Expression
alephsuc	|- (A e. On -> (aleph` suc A) = |^|{x e. On | (aleph` A) ~< x})

Distinct variable group(s):   x,A

Proof of Theorem alephsuc

Step	Hyp	Ref	Expression
1		fvex 2838	. . . 4 |- (aleph` A) e. V
2		numthcor 3601	. . . 4 |- ((aleph` A) e. V -> E.x e. On (aleph` A) ~< x)
3	1, 2	ax-mp 6	. . 3 |- E.x e. On (aleph` A) ~< x
4		intexrab 1988	. . 3 |- (E.x e. On (aleph` A) ~< x <-> |^|{x e. On | (aleph` A) ~< x} e. V)
5	3, 4	mpbi 164	. 2 |- |^|{x e. On | (aleph` A) ~< x} e. V
6		ax-17 925	. . 3 |- (w e. om -> A.y w e. om)
7		ax-17 925	. . 3 |- (w e. A -> A.y w e. A)
8		ax-17 925	. . 3 |- (w e. |^|{x e. On | (aleph` A) ~< x} -> A.y w e. |^|{x e. On | (aleph` A) ~< x})
9		df-aleph 3624	. . 3 |- aleph = rec({<.y, z>. | z = |^|{x e. On | y ~< x}}, om)
10		breq1 2065	. . . . 5 |- (y = (aleph` A) -> (y ~< x <-> (aleph` A) ~< x))
11	10	birabsdv 1344	. . . 4 |- (y = (aleph` A) -> {x e. On | y ~< x} = {x e. On | (aleph` A) ~< x})
12	11	inteqd 1970	. . 3 |- (y = (aleph` A) -> |^|{x e. On | y ~< x} = |^|{x e. On | (aleph` A) ~< x})
13	6, 7, 8, 9, 12	rdgsucopab 2984	. 2 |- ((A e. On /\ |^|{x e. On | (aleph` A) ~< x} e. V) -> (aleph` suc A) = |^|{x e. On | (aleph` A) ~< x})
14	5, 13	mpan2 519	1 |- (A e. On -> (aleph` suc A) = |^|{x e. On | (aleph` A) ~< x})

Syntax hints:   -> wi 2   = wceq 1091   e. wcel 1092  E.wrex 1202  {crab 1204  Vcvv 1348  |^|cint 1965   class class class wbr 2054  Oncon0 2199  suc csuc 2201  omcom 2372  ` cfv 2422   ~< csdm 3273  alephcale 3621

This theorem is referenced by:  alephcard 3673  alephnbtwn 3674  alephordlem1 3677  cardaleph 3690

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-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-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-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-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-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-aleph 3624

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