Theorem alephcard 4850

Description: Every aleph is a cardinal number. Theorem 65 of [Suppes] p. 229.

Assertion

Ref	Expression
alephcard	|- (card` (aleph` A)) = (aleph` A)

Proof of Theorem alephcard

Step	Hyp	Ref	Expression
1		fveq2 3719	. . . . 5 |- (x = (/) -> (aleph` x) = (aleph` (/)))
2	1	fveq2d 3723	. . . 4 |- (x = (/) -> (card` (aleph` x)) = (card` (aleph` (/))))
3	2, 1	eqeq12d 1487	. . 3 |- (x = (/) -> ((card` (aleph` x)) = (aleph` x) <-> (card` (aleph` (/))) = (aleph` (/))))
4		fveq2 3719	. . . . 5 |- (x = y -> (aleph` x) = (aleph` y))
5	4	fveq2d 3723	. . . 4 |- (x = y -> (card` (aleph` x)) = (card` (aleph` y)))
6	5, 4	eqeq12d 1487	. . 3 |- (x = y -> ((card` (aleph` x)) = (aleph` x) <-> (card` (aleph` y)) = (aleph` y)))
7		fveq2 3719	. . . . 5 |- (x = suc y -> (aleph` x) = (aleph` suc y))
8	7	fveq2d 3723	. . . 4 |- (x = suc y -> (card` (aleph` x)) = (card` (aleph` suc y)))
9	8, 7	eqeq12d 1487	. . 3 |- (x = suc y -> ((card` (aleph` x)) = (aleph` x) <-> (card` (aleph` suc y)) = (aleph` suc y)))
10		fveq2 3719	. . . . 5 |- (x = A -> (aleph` x) = (aleph` A))
11	10	fveq2d 3723	. . . 4 |- (x = A -> (card` (aleph` x)) = (card` (aleph` A)))
12	11, 10	eqeq12d 1487	. . 3 |- (x = A -> ((card` (aleph` x)) = (aleph` x) <-> (card` (aleph` A)) = (aleph` A)))
13		cardom 4808	. . . 4 |- (card` om) = om
14		aleph0 4846	. . . . 5 |- (aleph` (/)) = om
15	14	fveq2i 3722	. . . 4 |- (card` (aleph` (/))) = (card` om)
16	13, 15, 14	3eqtr4 1503	. . 3 |- (card` (aleph` (/))) = (aleph` (/))
17		fvex 3727	. . . . . 6 |- (aleph` y) e. V
18		cardmin 4843	. . . . . 6 |- ((aleph` y) e. V -> (card` |^|{x e. On | (aleph` y) ~< x}) = |^|{x e. On | (aleph` y) ~< x})
19	17, 18	ax-mp 7	. . . . 5 |- (card` |^|{x e. On | (aleph` y) ~< x}) = |^|{x e. On | (aleph` y) ~< x}
20		alephsuc 4849	. . . . . 6 |- (y e. On -> (aleph` suc y) = |^|{x e. On | (aleph` y) ~< x})
21	20	fveq2d 3723	. . . . 5 |- (y e. On -> (card` (aleph` suc y)) = (card` |^|{x e. On | (aleph` y) ~< x}))
22	19, 21, 20	3eqtr4a 1530	. . . 4 |- (y e. On -> (card` (aleph` suc y)) = (aleph` suc y))
23	22	a1d 12	. . 3 |- (y e. On -> ((card` (aleph` y)) = (aleph` y) -> (card` (aleph` suc y)) = (aleph` suc y)))
24		visset 1810	. . . . . . 7 |- x e. V
25		cardiun 4842	. . . . . . 7 |- (x e. V -> (A.y e. x (card` (aleph` y)) = (aleph` y) -> (card` U_y e. x (aleph` y)) = U_y e. x (aleph` y)))
26	24, 25	ax-mp 7	. . . . . 6 |- (A.y e. x (card` (aleph` y)) = (aleph` y) -> (card` U_y e. x (aleph` y)) = U_y e. x (aleph` y))
27	26	adantl 388	. . . . 5 |- ((Lim x /\ A.y e. x (card` (aleph` y)) = (aleph` y)) -> (card` U_y e. x (aleph` y)) = U_y e. x (aleph` y))
28		alephlim 4847	. . . . . . . 8 |- ((x e. V /\ Lim x) -> (aleph` x) = U_y e. x (aleph` y))
29	24, 28	mpan 694	. . . . . . 7 |- (Lim x -> (aleph` x) = U_y e. x (aleph` y))
30	29	adantr 389	. . . . . 6 |- ((Lim x /\ A.y e. x (card` (aleph` y)) = (aleph` y)) -> (aleph` x) = U_y e. x (aleph` y))
31	30	fveq2d 3723	. . . . 5 |- ((Lim x /\ A.y e. x (card` (aleph` y)) = (aleph` y)) -> (card` (aleph` x)) = (card` U_y e. x (aleph` y)))
32	27, 31, 30	3eqtr4d 1515	. . . 4 |- ((Lim x /\ A.y e. x (card` (aleph` y)) = (aleph` y)) -> (card` (aleph` x)) = (aleph` x))
33	32	ex 373	. . 3 |- (Lim x -> (A.y e. x (card` (aleph` y)) = (aleph` y) -> (card` (aleph` x)) = (aleph` x)))
34	3, 6, 9, 12, 16, 23, 33	tfinds 3157	. 2 |- (A e. On -> (card` (aleph` A)) = (aleph` A))
35		card0 4806	. . 3 |- (card` (/)) = (/)
36		alephfnon 4845	. . . . . . . 8 |- aleph Fn On
37		fndm 3583	. . . . . . . 8 |- (aleph Fn On -> dom aleph = On)
38	36, 37	ax-mp 7	. . . . . . 7 |- dom aleph = On
39	38	eleq2i 1536	. . . . . 6 |- (A e. dom aleph <-> A e. On)
40	39	negbii 187	. . . . 5 |- (-. A e. dom aleph <-> -. A e. On)
41		ndmfv 3740	. . . . 5 |- (-. A e. dom aleph -> (aleph` A) = (/))
42	40, 41	sylbir 201	. . . 4 |- (-. A e. On -> (aleph` A) = (/))
43	42	fveq2d 3723	. . 3 |- (-. A e. On -> (card` (aleph` A)) = (card` (/)))
44	35, 43, 42	3eqtr4a 1530	. 2 |- (-. A e. On -> (card` (aleph` A)) = (aleph` A))
45	34, 44	pm2.61i 126	1 |- (card` (aleph` A)) = (aleph` A)

Syntax hints:  -. wn 2   -> wi 3   /\ wa 223   = wceq 955   e. wcel 957  A.wral 1643  {crab 1646  Vcvv 1808  (/)c0 2277  |^|cint 2529  U_ciun 2562   class class class wbr 2615  Oncon0 2944  Lim wlim 2945  suc csuc 2946  omcom 3127  dom cdm 3166   Fn wfn 3173  ` cfv 3178   ~< csdm 4359  cardccrd 4796  alephcale 4797

This theorem is referenced by:  alephnbtwn2 4852  alephord2 4859  alephsuc2 4864  alephislim 4866  cardaleph 4868  cardalephex 4869  alephval2 4885  alephval3 4886  alephsuc3 7545

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-9 964  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458  ax-rep 2689  ax-sep 2699  ax-nul 2706  ax-pow 2738  ax-pr 2775  ax-un 2862  ax-reg 4576  ax-inf2 4608  ax-ac 4727

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 980  df-sb 1171  df-eu 1381  df-mo 1382  df-clab 1463  df-cleq 1468  df-clel 1471  df-ne 1585  df-ral 1647  df-rex 1648  df-reu 1649  df-rab 1650  df-v 1809  df-sbc 1939  df-dif 2046  df-un 2047  df-in 2048  df-ss 2050  df-pss 2052  df-nul 2278  df-if 2359  df-pw 2399  df-sn 2409  df-pr 2410  df-tp 2412  df-op 2413  df-uni 2500  df-int 2530  df-iun 2564  df-br 2616  df-opab 2663  df-tr 2677  df-eprel 2828  df-id 2831  df-po 2836  df-so 2846  df-fr 2913  df-we 2930  df-ord 2947  df-on 2948  df-lim 2949  df-suc 2950  df-om 3128  df-xp 3180  df-rel 3181  df-cnv 3182  df-co 3183  df-dm 3184  df-rn 3185  df-res 3186  df-ima 3187  df-fun 3188  df-fn 3189  df-f 3190  df-f1 3191  df-fo 3192  df-f1o 3193  df-fv 3194  df-rdg 3927  df-er 4254  df-en 4360  df-dom 4361  df-sdom 4362  df-card 4799  df-aleph 4800

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