Theorem alephval3 10032

Description: An alternate way to express the value of the aleph function: it is the least infinite cardinal different from all values at smaller arguments. Definition of aleph in [Enderton] p. 212 and definition of aleph in [BellMachover] p. 490 . (Contributed by NM, 16-Nov-2003.)

Assertion

Ref	Expression
alephval3	⊢ (𝐴 ∈ On → (ℵ‘𝐴) = ∩ {𝑥 ∣ ((card‘𝑥) = 𝑥 ∧ ω ⊆ 𝑥 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑥 = (ℵ‘𝑦))})

Distinct variable group:   𝑥,𝑦,𝐴

Proof of Theorem alephval3

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		alephcard 9992	. . . 4 ⊢ (card‘(ℵ‘𝐴)) = (ℵ‘𝐴)
2	1	a1i 11	. . 3 ⊢ (𝐴 ∈ On → (card‘(ℵ‘𝐴)) = (ℵ‘𝐴))
3		alephgeom 10004	. . . 4 ⊢ (𝐴 ∈ On ↔ ω ⊆ (ℵ‘𝐴))
4	3	biimpi 216	. . 3 ⊢ (𝐴 ∈ On → ω ⊆ (ℵ‘𝐴))
5		alephord2i 9999	. . . . 5 ⊢ (𝐴 ∈ On → (𝑦 ∈ 𝐴 → (ℵ‘𝑦) ∈ (ℵ‘𝐴)))
6		alephon 9991	. . . . . . . 8 ⊢ (ℵ‘𝑦) ∈ On
7	6	onirri 6438	. . . . . . 7 ⊢ ¬ (ℵ‘𝑦) ∈ (ℵ‘𝑦)
8		eleq2 2826	. . . . . . 7 ⊢ ((ℵ‘𝐴) = (ℵ‘𝑦) → ((ℵ‘𝑦) ∈ (ℵ‘𝐴) ↔ (ℵ‘𝑦) ∈ (ℵ‘𝑦)))
9	7, 8	mtbiri 327	. . . . . 6 ⊢ ((ℵ‘𝐴) = (ℵ‘𝑦) → ¬ (ℵ‘𝑦) ∈ (ℵ‘𝐴))
10	9	con2i 139	. . . . 5 ⊢ ((ℵ‘𝑦) ∈ (ℵ‘𝐴) → ¬ (ℵ‘𝐴) = (ℵ‘𝑦))
11	5, 10	syl6 35	. . . 4 ⊢ (𝐴 ∈ On → (𝑦 ∈ 𝐴 → ¬ (ℵ‘𝐴) = (ℵ‘𝑦)))
12	11	ralrimiv 3129	. . 3 ⊢ (𝐴 ∈ On → ∀𝑦 ∈ 𝐴 ¬ (ℵ‘𝐴) = (ℵ‘𝑦))
13		fvex 6854	. . . 4 ⊢ (ℵ‘𝐴) ∈ V
14		fveq2 6841	. . . . . 6 ⊢ (𝑥 = (ℵ‘𝐴) → (card‘𝑥) = (card‘(ℵ‘𝐴)))
15		id 22	. . . . . 6 ⊢ (𝑥 = (ℵ‘𝐴) → 𝑥 = (ℵ‘𝐴))
16	14, 15	eqeq12d 2753	. . . . 5 ⊢ (𝑥 = (ℵ‘𝐴) → ((card‘𝑥) = 𝑥 ↔ (card‘(ℵ‘𝐴)) = (ℵ‘𝐴)))
17		sseq2 3949	. . . . 5 ⊢ (𝑥 = (ℵ‘𝐴) → (ω ⊆ 𝑥 ↔ ω ⊆ (ℵ‘𝐴)))
18		eqeq1 2741	. . . . . . 7 ⊢ (𝑥 = (ℵ‘𝐴) → (𝑥 = (ℵ‘𝑦) ↔ (ℵ‘𝐴) = (ℵ‘𝑦)))
19	18	notbid 318	. . . . . 6 ⊢ (𝑥 = (ℵ‘𝐴) → (¬ 𝑥 = (ℵ‘𝑦) ↔ ¬ (ℵ‘𝐴) = (ℵ‘𝑦)))
20	19	ralbidv 3161	. . . . 5 ⊢ (𝑥 = (ℵ‘𝐴) → (∀𝑦 ∈ 𝐴 ¬ 𝑥 = (ℵ‘𝑦) ↔ ∀𝑦 ∈ 𝐴 ¬ (ℵ‘𝐴) = (ℵ‘𝑦)))
21	16, 17, 20	3anbi123d 1439	. . . 4 ⊢ (𝑥 = (ℵ‘𝐴) → (((card‘𝑥) = 𝑥 ∧ ω ⊆ 𝑥 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑥 = (ℵ‘𝑦)) ↔ ((card‘(ℵ‘𝐴)) = (ℵ‘𝐴) ∧ ω ⊆ (ℵ‘𝐴) ∧ ∀𝑦 ∈ 𝐴 ¬ (ℵ‘𝐴) = (ℵ‘𝑦))))
22	13, 21	elab 3623	. . 3 ⊢ ((ℵ‘𝐴) ∈ {𝑥 ∣ ((card‘𝑥) = 𝑥 ∧ ω ⊆ 𝑥 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑥 = (ℵ‘𝑦))} ↔ ((card‘(ℵ‘𝐴)) = (ℵ‘𝐴) ∧ ω ⊆ (ℵ‘𝐴) ∧ ∀𝑦 ∈ 𝐴 ¬ (ℵ‘𝐴) = (ℵ‘𝑦)))
23	2, 4, 12, 22	syl3anbrc 1345	. 2 ⊢ (𝐴 ∈ On → (ℵ‘𝐴) ∈ {𝑥 ∣ ((card‘𝑥) = 𝑥 ∧ ω ⊆ 𝑥 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑥 = (ℵ‘𝑦))})
24		eleq1 2825	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = (ℵ‘𝑦) → (𝑧 ∈ (ℵ‘𝐴) ↔ (ℵ‘𝑦) ∈ (ℵ‘𝐴)))
25		alephord2 9998	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ On ∧ 𝐴 ∈ On) → (𝑦 ∈ 𝐴 ↔ (ℵ‘𝑦) ∈ (ℵ‘𝐴)))
26	25	bicomd 223	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ On ∧ 𝐴 ∈ On) → ((ℵ‘𝑦) ∈ (ℵ‘𝐴) ↔ 𝑦 ∈ 𝐴))
27	24, 26	sylan9bbr 510	. . . . . . . . . . . . . 14 ⊢ (((𝑦 ∈ On ∧ 𝐴 ∈ On) ∧ 𝑧 = (ℵ‘𝑦)) → (𝑧 ∈ (ℵ‘𝐴) ↔ 𝑦 ∈ 𝐴))
28	27	biimpcd 249	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ (ℵ‘𝐴) → (((𝑦 ∈ On ∧ 𝐴 ∈ On) ∧ 𝑧 = (ℵ‘𝑦)) → 𝑦 ∈ 𝐴))
29		simpr 484	. . . . . . . . . . . . 13 ⊢ (((𝑦 ∈ On ∧ 𝐴 ∈ On) ∧ 𝑧 = (ℵ‘𝑦)) → 𝑧 = (ℵ‘𝑦))
30	28, 29	jca2 513	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ (ℵ‘𝐴) → (((𝑦 ∈ On ∧ 𝐴 ∈ On) ∧ 𝑧 = (ℵ‘𝑦)) → (𝑦 ∈ 𝐴 ∧ 𝑧 = (ℵ‘𝑦))))
31	30	exp4c 432	. . . . . . . . . . 11 ⊢ (𝑧 ∈ (ℵ‘𝐴) → (𝑦 ∈ On → (𝐴 ∈ On → (𝑧 = (ℵ‘𝑦) → (𝑦 ∈ 𝐴 ∧ 𝑧 = (ℵ‘𝑦))))))
32	31	com3r 87	. . . . . . . . . 10 ⊢ (𝐴 ∈ On → (𝑧 ∈ (ℵ‘𝐴) → (𝑦 ∈ On → (𝑧 = (ℵ‘𝑦) → (𝑦 ∈ 𝐴 ∧ 𝑧 = (ℵ‘𝑦))))))
33	32	imp4b 421	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ 𝑧 ∈ (ℵ‘𝐴)) → ((𝑦 ∈ On ∧ 𝑧 = (ℵ‘𝑦)) → (𝑦 ∈ 𝐴 ∧ 𝑧 = (ℵ‘𝑦))))
34	33	reximdv2 3148	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ 𝑧 ∈ (ℵ‘𝐴)) → (∃𝑦 ∈ On 𝑧 = (ℵ‘𝑦) → ∃𝑦 ∈ 𝐴 𝑧 = (ℵ‘𝑦)))
35		cardalephex 10012	. . . . . . . . 9 ⊢ (ω ⊆ 𝑧 → ((card‘𝑧) = 𝑧 ↔ ∃𝑦 ∈ On 𝑧 = (ℵ‘𝑦)))
36	35	biimpac 478	. . . . . . . 8 ⊢ (((card‘𝑧) = 𝑧 ∧ ω ⊆ 𝑧) → ∃𝑦 ∈ On 𝑧 = (ℵ‘𝑦))
37	34, 36	impel 505	. . . . . . 7 ⊢ (((𝐴 ∈ On ∧ 𝑧 ∈ (ℵ‘𝐴)) ∧ ((card‘𝑧) = 𝑧 ∧ ω ⊆ 𝑧)) → ∃𝑦 ∈ 𝐴 𝑧 = (ℵ‘𝑦))
38		dfrex2 3065	. . . . . . 7 ⊢ (∃𝑦 ∈ 𝐴 𝑧 = (ℵ‘𝑦) ↔ ¬ ∀𝑦 ∈ 𝐴 ¬ 𝑧 = (ℵ‘𝑦))
39	37, 38	sylib 218	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ 𝑧 ∈ (ℵ‘𝐴)) ∧ ((card‘𝑧) = 𝑧 ∧ ω ⊆ 𝑧)) → ¬ ∀𝑦 ∈ 𝐴 ¬ 𝑧 = (ℵ‘𝑦))
40		nan 830	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ 𝑧 ∈ (ℵ‘𝐴)) → ¬ (((card‘𝑧) = 𝑧 ∧ ω ⊆ 𝑧) ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑧 = (ℵ‘𝑦))) ↔ (((𝐴 ∈ On ∧ 𝑧 ∈ (ℵ‘𝐴)) ∧ ((card‘𝑧) = 𝑧 ∧ ω ⊆ 𝑧)) → ¬ ∀𝑦 ∈ 𝐴 ¬ 𝑧 = (ℵ‘𝑦)))
41	39, 40	mpbir 231	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝑧 ∈ (ℵ‘𝐴)) → ¬ (((card‘𝑧) = 𝑧 ∧ ω ⊆ 𝑧) ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑧 = (ℵ‘𝑦)))
42	41	ex 412	. . . 4 ⊢ (𝐴 ∈ On → (𝑧 ∈ (ℵ‘𝐴) → ¬ (((card‘𝑧) = 𝑧 ∧ ω ⊆ 𝑧) ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑧 = (ℵ‘𝑦))))
43		vex 3434	. . . . . . 7 ⊢ 𝑧 ∈ V
44		fveq2 6841	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → (card‘𝑥) = (card‘𝑧))
45		id 22	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → 𝑥 = 𝑧)
46	44, 45	eqeq12d 2753	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → ((card‘𝑥) = 𝑥 ↔ (card‘𝑧) = 𝑧))
47		sseq2 3949	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (ω ⊆ 𝑥 ↔ ω ⊆ 𝑧))
48		eqeq1 2741	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → (𝑥 = (ℵ‘𝑦) ↔ 𝑧 = (ℵ‘𝑦)))
49	48	notbid 318	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → (¬ 𝑥 = (ℵ‘𝑦) ↔ ¬ 𝑧 = (ℵ‘𝑦)))
50	49	ralbidv 3161	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (∀𝑦 ∈ 𝐴 ¬ 𝑥 = (ℵ‘𝑦) ↔ ∀𝑦 ∈ 𝐴 ¬ 𝑧 = (ℵ‘𝑦)))
51	46, 47, 50	3anbi123d 1439	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (((card‘𝑥) = 𝑥 ∧ ω ⊆ 𝑥 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑥 = (ℵ‘𝑦)) ↔ ((card‘𝑧) = 𝑧 ∧ ω ⊆ 𝑧 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑧 = (ℵ‘𝑦))))
52	43, 51	elab 3623	. . . . . 6 ⊢ (𝑧 ∈ {𝑥 ∣ ((card‘𝑥) = 𝑥 ∧ ω ⊆ 𝑥 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑥 = (ℵ‘𝑦))} ↔ ((card‘𝑧) = 𝑧 ∧ ω ⊆ 𝑧 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑧 = (ℵ‘𝑦)))
53		df-3an 1089	. . . . . 6 ⊢ (((card‘𝑧) = 𝑧 ∧ ω ⊆ 𝑧 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑧 = (ℵ‘𝑦)) ↔ (((card‘𝑧) = 𝑧 ∧ ω ⊆ 𝑧) ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑧 = (ℵ‘𝑦)))
54	52, 53	bitri 275	. . . . 5 ⊢ (𝑧 ∈ {𝑥 ∣ ((card‘𝑥) = 𝑥 ∧ ω ⊆ 𝑥 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑥 = (ℵ‘𝑦))} ↔ (((card‘𝑧) = 𝑧 ∧ ω ⊆ 𝑧) ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑧 = (ℵ‘𝑦)))
55	54	notbii 320	. . . 4 ⊢ (¬ 𝑧 ∈ {𝑥 ∣ ((card‘𝑥) = 𝑥 ∧ ω ⊆ 𝑥 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑥 = (ℵ‘𝑦))} ↔ ¬ (((card‘𝑧) = 𝑧 ∧ ω ⊆ 𝑧) ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑧 = (ℵ‘𝑦)))
56	42, 55	imbitrrdi 252	. . 3 ⊢ (𝐴 ∈ On → (𝑧 ∈ (ℵ‘𝐴) → ¬ 𝑧 ∈ {𝑥 ∣ ((card‘𝑥) = 𝑥 ∧ ω ⊆ 𝑥 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑥 = (ℵ‘𝑦))}))
57	56	ralrimiv 3129	. 2 ⊢ (𝐴 ∈ On → ∀𝑧 ∈ (ℵ‘𝐴) ¬ 𝑧 ∈ {𝑥 ∣ ((card‘𝑥) = 𝑥 ∧ ω ⊆ 𝑥 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑥 = (ℵ‘𝑦))})
58		cardon 9868	. . . . . 6 ⊢ (card‘𝑥) ∈ On
59		eleq1 2825	. . . . . 6 ⊢ ((card‘𝑥) = 𝑥 → ((card‘𝑥) ∈ On ↔ 𝑥 ∈ On))
60	58, 59	mpbii 233	. . . . 5 ⊢ ((card‘𝑥) = 𝑥 → 𝑥 ∈ On)
61	60	3ad2ant1 1134	. . . 4 ⊢ (((card‘𝑥) = 𝑥 ∧ ω ⊆ 𝑥 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑥 = (ℵ‘𝑦)) → 𝑥 ∈ On)
62	61	abssi 4009	. . 3 ⊢ {𝑥 ∣ ((card‘𝑥) = 𝑥 ∧ ω ⊆ 𝑥 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑥 = (ℵ‘𝑦))} ⊆ On
63		oneqmini 6377	. . 3 ⊢ ({𝑥 ∣ ((card‘𝑥) = 𝑥 ∧ ω ⊆ 𝑥 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑥 = (ℵ‘𝑦))} ⊆ On → (((ℵ‘𝐴) ∈ {𝑥 ∣ ((card‘𝑥) = 𝑥 ∧ ω ⊆ 𝑥 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑥 = (ℵ‘𝑦))} ∧ ∀𝑧 ∈ (ℵ‘𝐴) ¬ 𝑧 ∈ {𝑥 ∣ ((card‘𝑥) = 𝑥 ∧ ω ⊆ 𝑥 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑥 = (ℵ‘𝑦))}) → (ℵ‘𝐴) = ∩ {𝑥 ∣ ((card‘𝑥) = 𝑥 ∧ ω ⊆ 𝑥 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑥 = (ℵ‘𝑦))}))
64	62, 63	ax-mp 5	. 2 ⊢ (((ℵ‘𝐴) ∈ {𝑥 ∣ ((card‘𝑥) = 𝑥 ∧ ω ⊆ 𝑥 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑥 = (ℵ‘𝑦))} ∧ ∀𝑧 ∈ (ℵ‘𝐴) ¬ 𝑧 ∈ {𝑥 ∣ ((card‘𝑥) = 𝑥 ∧ ω ⊆ 𝑥 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑥 = (ℵ‘𝑦))}) → (ℵ‘𝐴) = ∩ {𝑥 ∣ ((card‘𝑥) = 𝑥 ∧ ω ⊆ 𝑥 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑥 = (ℵ‘𝑦))})
65	23, 57, 64	syl2anc 585	1 ⊢ (𝐴 ∈ On → (ℵ‘𝐴) = ∩ {𝑥 ∣ ((card‘𝑥) = 𝑥 ∧ ω ⊆ 𝑥 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑥 = (ℵ‘𝑦))})

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  {cab 2715  ∀wral 3052  ∃wrex 3062   ⊆ wss 3890  ∩ cint 4890  Oncon0 6324  ‘cfv 6499  ωcom 7817  cardccrd 9859  ℵcale 9860

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pow 5308  ax-pr 5376  ax-un 7689  ax-inf2 9562

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-se 5585  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6266  df-ord 6327  df-on 6328  df-lim 6329  df-suc 6330  df-iota 6455  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-isom 6508  df-riota 7324  df-ov 7370  df-om 7818  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-1o 8405  df-er 8643  df-en 8894  df-dom 8895  df-sdom 8896  df-fin 8897  df-oi 9425  df-har 9472  df-card 9863  df-aleph 9864

This theorem is referenced by: (None)