Theorem alephcard 4867

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 3724	. . . . 5 |- (x = (/) -> (aleph` x) = (aleph` (/)))
2	1	fveq2d 3728	. . . 4 |- (x = (/) -> (card` (aleph` x)) = (card` (aleph` (/))))
3	2, 1	eqeq12d 1489	. . 3 |- (x = (/) -> ((card` (aleph` x)) = (aleph` x) <-> (card` (aleph` (/))) = (aleph` (/))))
4		fveq2 3724	. . . . 5 |- (x = y -> (aleph` x) = (aleph` y))
5	4	fveq2d 3728	. . . 4 |- (x = y -> (card` (aleph` x)) = (card` (aleph` y)))
6	5, 4	eqeq12d 1489	. . 3 |- (x = y -> ((card` (aleph` x)) = (aleph` x) <-> (card` (aleph` y)) = (aleph` y)))
7		fveq2 3724	. . . . 5 |- (x = suc y -> (aleph` x) = (aleph` suc y))
8	7	fveq2d 3728	. . . 4 |- (x = suc y -> (card` (aleph` x)) = (card` (aleph` suc y)))
9	8, 7	eqeq12d 1489	. . 3 |- (x = suc y -> ((card` (aleph` x)) = (aleph` x) <-> (card` (aleph` suc y)) = (aleph` suc y)))
10		fveq2 3724	. . . . 5 |- (x = A -> (aleph` x) = (aleph` A))
11	10	fveq2d 3728	. . . 4 |- (x = A -> (card` (aleph` x)) = (card` (aleph` A)))
12	11, 10	eqeq12d 1489	. . 3 |- (x = A -> ((card` (aleph` x)) = (aleph` x) <-> (card` (aleph` A)) = (aleph` A)))
13		cardom 4825	. . . 4 |- (card` om) = om
14		aleph0 4863	. . . . 5 |- (aleph` (/)) = om
15	14	fveq2i 3727	. . . 4 |- (card` (aleph` (/))) = (card` om)
16	13, 15, 14	3eqtr4 1505	. . 3 |- (card` (aleph` (/))) = (aleph` (/))
17		fvex 3732	. . . . . 6 |- (aleph` y) e. V
18		cardmin 4860	. . . . . 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 4866	. . . . . 6 |- (y e. On -> (aleph` suc y) = |^|{x e. On | (aleph` y) ~< x})
21	20	fveq2d 3728	. . . . 5 |- (y e. On -> (card` (aleph` suc y)) = (card` |^|{x e. On | (aleph` y) ~< x}))
22	19, 21, 20	3eqtr4a 1532	. . . 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 1813	. . . . . . 7 |- x e. V
25		cardiun 4859	. . . . . . 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 4864	. . . . . . . 8 |- ((x e. V /\ Lim x) -> (aleph` x) = U_y e. x (aleph` y))
29	24, 28	mpan 695	. . . . . . 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 3728	. . . . 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 1517	. . . 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 3161	. 2 |- (A e. On -> (card` (aleph` A)) = (aleph` A))
35		card0 4823	. . 3 |- (card` (/)) = (/)
36		alephfnon 4862	. . . . . . . 8 |- aleph Fn On
37		fndm 3587	. . . . . . . 8 |- (aleph Fn On -> dom aleph = On)
38	36, 37	ax-mp 7	. . . . . . 7 |- dom aleph = On
39	38	eleq2i 1538	. . . . . 6 |- (A e. dom aleph <-> A e. On)
40	39	negbii 187	. . . . 5 |- (-. A e. dom aleph <-> -. A e. On)
41		ndmfv 3745	. . . . 5 |- (-. A e. dom aleph -> (aleph` A) = (/))
42	40, 41	sylbir 201	. . . 4 |- (-. A e. On -> (aleph` A) = (/))
43	42	fveq2d 3728	. . 3 |- (-. A e. On -> (card` (aleph` A)) = (card` (/)))
44	35, 43, 42	3eqtr4a 1532	. 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 956   e. wcel 958  A.wral 1645  {crab 1648  Vcvv 1811  (/)c0 2280  |^|cint 2533  U_ciun 2566   class class class wbr 2619  Oncon0 2948  Lim wlim 2949  suc csuc 2950  omcom 3131  dom cdm 3170   Fn wfn 3177  ` cfv 3182   ~< csdm 4366  cardccrd 4813  alephcale 4814

This theorem is referenced by:  alephnbtwn2 4869  alephord2 4876  alephsuc2 4881  alephislim 4883  cardaleph 4885  cardalephex 4886  alephval2 4902  alephval3 4903  alephsuc3 7585

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2693  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866  ax-reg 4593  ax-inf2 4625  ax-ac 4744

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 776  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-reu 1651  df-rab 1652  df-v 1812  df-sbc 1942  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-pss 2055  df-nul 2281  df-if 2362  df-pw 2402  df-sn 2412  df-pr 2413  df-tp 2415  df-op 2416  df-uni 2504  df-int 2534  df-iun 2568  df-br 2620  df-opab 2667  df-tr 2681  df-eprel 2832  df-id 2835  df-po 2840  df-so 2850  df-fr 2917  df-we 2934  df-ord 2951  df-on 2952  df-lim 2953  df-suc 2954  df-om 3132  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-f 3194  df-f1 3195  df-fo 3196  df-f1o 3197  df-fv 3198  df-rdg 3932  df-er 4261  df-en 4368  df-dom 4369  df-sdom 4370  df-card 4816  df-aleph 4817

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