Theorem alephsuc 4877

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:   x,A

Proof of Theorem alephsuc

Step	Hyp	Ref	Expression
1		fvex 3738	. . . 4 |- (aleph` A) e. V
2		numthcor 4796	. . . 4 |- ((aleph` A) e. V -> E.x e. On (aleph` A) ~< x)
3	1, 2	ax-mp 7	. . 3 |- E.x e. On (aleph` A) ~< x
4		intexrab 2737	. . 3 |- (E.x e. On (aleph` A) ~< x <-> |^|{x e. On | (aleph` A) ~< x} e. V)
5	3, 4	mpbi 189	. 2 |- |^|{x e. On | (aleph` A) ~< x} e. V
6		ax-17 973	. . 3 |- (w e. om -> A.y w e. om)
7		ax-17 973	. . 3 |- (w e. A -> A.y w e. A)
8		ax-17 973	. . 3 |- (w e. |^|{x e. On | (aleph` A) ~< x} -> A.y w e. |^|{x e. On | (aleph` A) ~< x})
9		df-aleph 4827	. . 3 |- aleph = rec({<.y, z>. | z = |^|{x e. On | y ~< x}}, om)
10		breq1 2627	. . . . 5 |- (y = (aleph` A) -> (y ~< x <-> (aleph` A) ~< x))
11	10	rabbisdv 1810	. . . 4 |- (y = (aleph` A) -> {x e. On | y ~< x} = {x e. On | (aleph` A) ~< x})
12	11	inteqd 2542	. . 3 |- (y = (aleph` A) -> |^|{x e. On | y ~< x} = |^|{x e. On | (aleph` A) ~< x})
13	6, 7, 8, 9, 12	rdgsucopab 3952	. 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 698	1 |- (A e. On -> (aleph` suc A) = |^|{x e. On | (aleph` A) ~< x})

Syntax hints:   -> wi 3   = wceq 958   e. wcel 960  E.wrex 1649  {crab 1651  Vcvv 1814  |^|cint 2537   class class class wbr 2624  Oncon0 2954  suc csuc 2956  omcom 3137  ` cfv 3188   ~< csdm 4372  alephcale 4824

This theorem is referenced by:  alephcard 4878  alephnbtwn 4879  alephordlem1 4883  cardaleph 4896

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-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-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-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-aleph 4827

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