Theorem alephcard 3673

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

Assertion

Ref	Expression
alephcard	⊢ (card ‘(ℵ ‘A)) = (ℵ ‘A)

Proof of Theorem alephcard

Step	Hyp	Ref	Expression
1		fveq2 2832	. . . . 5 ⊢ (x = ∅ → (ℵ ‘x) = (ℵ ‘∅))
2	1	fveq2d 2836	. . . 4 ⊢ (x = ∅ → (card ‘(ℵ ‘x)) = (card ‘(ℵ ‘∅)))
3	2, 1	cleq12d 1115	. . 3 ⊢ (x = ∅ → ((card ‘(ℵ ‘x)) = (ℵ ‘x) ↔ (card ‘(ℵ ‘∅)) = (ℵ ‘∅)))
4		fveq2 2832	. . . . 5 ⊢ (x = y → (ℵ ‘x) = (ℵ ‘y))
5	4	fveq2d 2836	. . . 4 ⊢ (x = y → (card ‘(ℵ ‘x)) = (card ‘(ℵ ‘y)))
6	5, 4	cleq12d 1115	. . 3 ⊢ (x = y → ((card ‘(ℵ ‘x)) = (ℵ ‘x) ↔ (card ‘(ℵ ‘y)) = (ℵ ‘y)))
7		fveq2 2832	. . . . 5 ⊢ (x = suc y → (ℵ ‘x) = (ℵ ‘suc y))
8	7	fveq2d 2836	. . . 4 ⊢ (x = suc y → (card ‘(ℵ ‘x)) = (card ‘(ℵ ‘suc y)))
9	8, 7	cleq12d 1115	. . 3 ⊢ (x = suc y → ((card ‘(ℵ ‘x)) = (ℵ ‘x) ↔ (card ‘(ℵ ‘suc y)) = (ℵ ‘suc y)))
10		fveq2 2832	. . . . 5 ⊢ (x = A → (ℵ ‘x) = (ℵ ‘A))
11	10	fveq2d 2836	. . . 4 ⊢ (x = A → (card ‘(ℵ ‘x)) = (card ‘(ℵ ‘A)))
12	11, 10	cleq12d 1115	. . 3 ⊢ (x = A → ((card ‘(ℵ ‘x)) = (ℵ ‘x) ↔ (card ‘(ℵ ‘A)) = (ℵ ‘A)))
13		cardom 3632	. . . 4 ⊢ (card ‘ω) = ω
14		aleph0 3669	. . . . 5 ⊢ (ℵ ‘∅) = ω
15	14	fveq2i 2835	. . . 4 ⊢ (card ‘(ℵ ‘∅)) = (card ‘ω)
16	13, 15, 14	3eqtr4 1126	. . 3 ⊢ (card ‘(ℵ ‘∅)) = (ℵ ‘∅)
17		fvex 2838	. . . . . . 7 ⊢ (ℵ ‘y) ∈ V
18		cardmin 3666	. . . . . . 7 ⊢ ((ℵ ‘y) ∈ V → (card ‘∩{x ∈ On∣(ℵ ‘y) ≺ x}) = ∩{x ∈ On∣(ℵ ‘y) ≺ x})
19	17, 18	ax-mp 6	. . . . . 6 ⊢ (card ‘∩{x ∈ On∣(ℵ ‘y) ≺ x}) = ∩{x ∈ On∣(ℵ ‘y) ≺ x}
20	19	a1i 7	. . . . 5 ⊢ (y ∈ On → (card ‘∩{x ∈ On∣(ℵ ‘y) ≺ x}) = ∩{x ∈ On∣(ℵ ‘y) ≺ x})
21		alephsuc 3672	. . . . . 6 ⊢ (y ∈ On → (ℵ ‘suc y) = ∩{x ∈ On∣(ℵ ‘y) ≺ x})
22	21	fveq2d 2836	. . . . 5 ⊢ (y ∈ On → (card ‘(ℵ ‘suc y)) = (card ‘∩{x ∈ On∣(ℵ ‘y) ≺ x}))
23	20, 22, 21	3eqtr4d 1134	. . . 4 ⊢ (y ∈ On → (card ‘(ℵ ‘suc y)) = (ℵ ‘suc y))
24	23	a1d 14	. . 3 ⊢ (y ∈ On → ((card ‘(ℵ ‘y)) = (ℵ ‘y) → (card ‘(ℵ ‘suc y)) = (ℵ ‘suc y)))
25		visset 1350	. . . . . . 7 ⊢ x ∈ V
26		cardiun 3665	. . . . . . 7 ⊢ (x ∈ V → (∀y ∈ x (card ‘(ℵ ‘y)) = (ℵ ‘y) → (card ‘∪y ∈ x (ℵ ‘y)) = ∪y ∈ x (ℵ ‘y)))
27	25, 26	ax-mp 6	. . . . . 6 ⊢ (∀y ∈ x (card ‘(ℵ ‘y)) = (ℵ ‘y) → (card ‘∪y ∈ x (ℵ ‘y)) = ∪y ∈ x (ℵ ‘y))
28	27	adantl 305	. . . . 5 ⊢ ((Lim x ∧ ∀y ∈ x (card ‘(ℵ ‘y)) = (ℵ ‘y)) → (card ‘∪y ∈ x (ℵ ‘y)) = ∪y ∈ x (ℵ ‘y))
29		alephlim 3670	. . . . . . . 8 ⊢ ((x ∈ V ∧ Lim x) → (ℵ ‘x) = ∪y ∈ x (ℵ ‘y))
30	25, 29	mpan 518	. . . . . . 7 ⊢ (Lim x → (ℵ ‘x) = ∪y ∈ x (ℵ ‘y))
31	30	adantr 306	. . . . . 6 ⊢ ((Lim x ∧ ∀y ∈ x (card ‘(ℵ ‘y)) = (ℵ ‘y)) → (ℵ ‘x) = ∪y ∈ x (ℵ ‘y))
32	31	fveq2d 2836	. . . . 5 ⊢ ((Lim x ∧ ∀y ∈ x (card ‘(ℵ ‘y)) = (ℵ ‘y)) → (card ‘(ℵ ‘x)) = (card ‘∪y ∈ x (ℵ ‘y)))
33	28, 32, 31	3eqtr4d 1134	. . . 4 ⊢ ((Lim x ∧ ∀y ∈ x (card ‘(ℵ ‘y)) = (ℵ ‘y)) → (card ‘(ℵ ‘x)) = (ℵ ‘x))
34	33	exp 291	. . 3 ⊢ (Lim x → (∀y ∈ x (card ‘(ℵ ‘y)) = (ℵ ‘y) → (card ‘(ℵ ‘x)) = (ℵ ‘x)))
35	3, 6, 9, 12, 16, 24, 34	tfinds 2401	. 2 ⊢ (A ∈ On → (card ‘(ℵ ‘A)) = (ℵ ‘A))
36		card0 3630	. . 3 ⊢ (card ‘∅) = ∅
37		alephfnon 3668	. . . . . . . . 9 ⊢ ℵ Fn On
38		fndm 2723	. . . . . . . . 9 ⊢ (ℵ Fn On → dom ℵ = On)
39	37, 38	ax-mp 6	. . . . . . . 8 ⊢ dom ℵ = On
40	39	eleq2i 1153	. . . . . . 7 ⊢ (A ∈ dom ℵ ↔ A ∈ On)
41	40	negbii 162	. . . . . 6 ⊢ (¬ A ∈ dom ℵ ↔ ¬ A ∈ On)
42		ndmfv 2848	. . . . . 6 ⊢ (¬ A ∈ dom ℵ → (ℵ ‘A) = ∅)
43	41, 42	sylbir 176	. . . . 5 ⊢ (¬ A ∈ On → (ℵ ‘A) = ∅)
44	43	fveq2d 2836	. . . 4 ⊢ (¬ A ∈ On → (card ‘(ℵ ‘A)) = (card ‘∅))
45	44, 43	cleq12d 1115	. . 3 ⊢ (¬ A ∈ On → ((card ‘(ℵ ‘A)) = (ℵ ‘A) ↔ (card ‘∅) = ∅))
46	36, 45	mpbiri 169	. 2 ⊢ (¬ A ∈ On → (card ‘(ℵ ‘A)) = (ℵ ‘A))
47	35, 46	pm2.61i 110	1 ⊢ (card ‘(ℵ ‘A)) = (ℵ ‘A)

Syntax hints:  ¬ wn 1   → wi 2   ∧ wa 196   = weq 797   = wceq 1091   ∈ wcel 1092  ∀wral 1201  {crab 1204  Vcvv 1348  ∅c0 1707  ∩cint 1965  ∪ciun 1994   class class class wbr 2054  Oncon0 2199  Lim wlim 2200  suc csuc 2201  ωcom 2372  dom cdm 2410   Fn wfn 2417   ‘cfv 2422   ≺ csdm 3273  cardccrd 3620  ℵcale 3621

This theorem is referenced by:  alephnbtwn2 3675  alephord2 3681  alephsuc2 3686  alephislim 3688  cardaleph 3690  cardalephex 3691  alephsuc3 4955

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-er 3200  df-en 3274  df-dom 3275  df-sdom 3276  df-card 3623  df-aleph 3624

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