Theorem alephval3 10216

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 10176	. . . 4 ⊢ (card‘(ℵ‘𝐴)) = (ℵ‘𝐴)
2	1	a1i 11	. . 3 ⊢ (𝐴 ∈ On → (card‘(ℵ‘𝐴)) = (ℵ‘𝐴))
3		alephgeom 10188	. . . 4 ⊢ (𝐴 ∈ On ↔ ω ⊆ (ℵ‘𝐴))
4	3	biimpi 219	. . 3 ⊢ (𝐴 ∈ On → ω ⊆ (ℵ‘𝐴))
5		alephord2i 10183	. . . . 5 ⊢ (𝐴 ∈ On → (𝑦 ∈ 𝐴 → (ℵ‘𝑦) ∈ (ℵ‘𝐴)))
6		alephon 10175	. . . . . . . 8 ⊢ (ℵ‘𝑦) ∈ On
7	6	onirri 6495	. . . . . . 7 ⊢ ¬ (ℵ‘𝑦) ∈ (ℵ‘𝑦)
8		eleq2 2853	. . . . . . 7 ⊢ ((ℵ‘𝐴) = (ℵ‘𝑦) → ((ℵ‘𝑦) ∈ (ℵ‘𝐴) ↔ (ℵ‘𝑦) ∈ (ℵ‘𝑦)))
9	7, 8	mtbiri 330	. . . . . 6 ⊢ ((ℵ‘𝐴) = (ℵ‘𝑦) → ¬ (ℵ‘𝑦) ∈ (ℵ‘𝐴))
10	9	con2i 140	. . . . 5 ⊢ ((ℵ‘𝑦) ∈ (ℵ‘𝐴) → ¬ (ℵ‘𝐴) = (ℵ‘𝑦))
11	5, 10	syl6 36	. . . 4 ⊢ (𝐴 ∈ On → (𝑦 ∈ 𝐴 → ¬ (ℵ‘𝐴) = (ℵ‘𝑦)))
12	11	ralrimiv 3157	. . 3 ⊢ (𝐴 ∈ On → ∀𝑦 ∈ 𝐴 ¬ (ℵ‘𝐴) = (ℵ‘𝑦))
13		fvex 6921	. . . 4 ⊢ (ℵ‘𝐴) ∈ V
14		fveq2 6908	. . . . . 6 ⊢ (𝑥 = (ℵ‘𝐴) → (card‘𝑥) = (card‘(ℵ‘𝐴)))
15		id 23	. . . . . 6 ⊢ (𝑥 = (ℵ‘𝐴) → 𝑥 = (ℵ‘𝐴))
16	14, 15	eqeq12d 2779	. . . . 5 ⊢ (𝑥 = (ℵ‘𝐴) → ((card‘𝑥) = 𝑥 ↔ (card‘(ℵ‘𝐴)) = (ℵ‘𝐴)))
17		sseq2 3939	. . . . 5 ⊢ (𝑥 = (ℵ‘𝐴) → (ω ⊆ 𝑥 ↔ ω ⊆ (ℵ‘𝐴)))
18		eqeq1 2767	. . . . . . 7 ⊢ (𝑥 = (ℵ‘𝐴) → (𝑥 = (ℵ‘𝑦) ↔ (ℵ‘𝐴) = (ℵ‘𝑦)))
19	18	notbid 321	. . . . . 6 ⊢ (𝑥 = (ℵ‘𝐴) → (¬ 𝑥 = (ℵ‘𝑦) ↔ ¬ (ℵ‘𝐴) = (ℵ‘𝑦)))
20	19	ralbidv 3189	. . . . 5 ⊢ (𝑥 = (ℵ‘𝐴) → (∀𝑦 ∈ 𝐴 ¬ 𝑥 = (ℵ‘𝑦) ↔ ∀𝑦 ∈ 𝐴 ¬ (ℵ‘𝐴) = (ℵ‘𝑦)))
21	16, 17, 20	3anbi123d 1465	. . . 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 3636	. . 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 1362	. 2 ⊢ (𝐴 ∈ On → (ℵ‘𝐴) ∈ {𝑥 ∣ ((card‘𝑥) = 𝑥 ∧ ω ⊆ 𝑥 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑥 = (ℵ‘𝑦))})
24		eleq1 2852	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = (ℵ‘𝑦) → (𝑧 ∈ (ℵ‘𝐴) ↔ (ℵ‘𝑦) ∈ (ℵ‘𝐴)))
25		alephord2 10182	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ On ∧ 𝐴 ∈ On) → (𝑦 ∈ 𝐴 ↔ (ℵ‘𝑦) ∈ (ℵ‘𝐴)))
26	25	bicomd 226	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ On ∧ 𝐴 ∈ On) → ((ℵ‘𝑦) ∈ (ℵ‘𝐴) ↔ 𝑦 ∈ 𝐴))
27	24, 26	sylan9bbr 520	. . . . . . . . . . . . . 14 ⊢ (((𝑦 ∈ On ∧ 𝐴 ∈ On) ∧ 𝑧 = (ℵ‘𝑦)) → (𝑧 ∈ (ℵ‘𝐴) ↔ 𝑦 ∈ 𝐴))
28	27	biimpcd 252	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ (ℵ‘𝐴) → (((𝑦 ∈ On ∧ 𝐴 ∈ On) ∧ 𝑧 = (ℵ‘𝑦)) → 𝑦 ∈ 𝐴))
29		simpr 490	. . . . . . . . . . . . 13 ⊢ (((𝑦 ∈ On ∧ 𝐴 ∈ On) ∧ 𝑧 = (ℵ‘𝑦)) → 𝑧 = (ℵ‘𝑦))
30	28, 29	jca2 523	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ (ℵ‘𝐴) → (((𝑦 ∈ On ∧ 𝐴 ∈ On) ∧ 𝑧 = (ℵ‘𝑦)) → (𝑦 ∈ 𝐴 ∧ 𝑧 = (ℵ‘𝑦))))
31	30	exp4c 438	. . . . . . . . . . 11 ⊢ (𝑧 ∈ (ℵ‘𝐴) → (𝑦 ∈ On → (𝐴 ∈ On → (𝑧 = (ℵ‘𝑦) → (𝑦 ∈ 𝐴 ∧ 𝑧 = (ℵ‘𝑦))))))
32	31	com3r 88	. . . . . . . . . 10 ⊢ (𝐴 ∈ On → (𝑧 ∈ (ℵ‘𝐴) → (𝑦 ∈ On → (𝑧 = (ℵ‘𝑦) → (𝑦 ∈ 𝐴 ∧ 𝑧 = (ℵ‘𝑦))))))
33	32	imp4b 427	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ 𝑧 ∈ (ℵ‘𝐴)) → ((𝑦 ∈ On ∧ 𝑧 = (ℵ‘𝑦)) → (𝑦 ∈ 𝐴 ∧ 𝑧 = (ℵ‘𝑦))))
34	33	reximdv2 3176	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ 𝑧 ∈ (ℵ‘𝐴)) → (∃𝑦 ∈ On 𝑧 = (ℵ‘𝑦) → ∃𝑦 ∈ 𝐴 𝑧 = (ℵ‘𝑦)))
35		cardalephex 10196	. . . . . . . . 9 ⊢ (ω ⊆ 𝑧 → ((card‘𝑧) = 𝑧 ↔ ∃𝑦 ∈ On 𝑧 = (ℵ‘𝑦)))
36	35	biimpac 484	. . . . . . . 8 ⊢ (((card‘𝑧) = 𝑧 ∧ ω ⊆ 𝑧) → ∃𝑦 ∈ On 𝑧 = (ℵ‘𝑦))
37	34, 36	impel 515	. . . . . . 7 ⊢ (((𝐴 ∈ On ∧ 𝑧 ∈ (ℵ‘𝐴)) ∧ ((card‘𝑧) = 𝑧 ∧ ω ⊆ 𝑧)) → ∃𝑦 ∈ 𝐴 𝑧 = (ℵ‘𝑦))
38		dfrex2 3093	. . . . . . 7 ⊢ (∃𝑦 ∈ 𝐴 𝑧 = (ℵ‘𝑦) ↔ ¬ ∀𝑦 ∈ 𝐴 ¬ 𝑧 = (ℵ‘𝑦))
39	37, 38	sylib 221	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ 𝑧 ∈ (ℵ‘𝐴)) ∧ ((card‘𝑧) = 𝑧 ∧ ω ⊆ 𝑧)) → ¬ ∀𝑦 ∈ 𝐴 ¬ 𝑧 = (ℵ‘𝑦))
40		nan 843	. . . . . 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 234	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝑧 ∈ (ℵ‘𝐴)) → ¬ (((card‘𝑧) = 𝑧 ∧ ω ⊆ 𝑧) ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑧 = (ℵ‘𝑦)))
42	41	ex 418	. . . 4 ⊢ (𝐴 ∈ On → (𝑧 ∈ (ℵ‘𝐴) → ¬ (((card‘𝑧) = 𝑧 ∧ ω ⊆ 𝑧) ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑧 = (ℵ‘𝑦))))
43		vex 3458	. . . . . . 7 ⊢ 𝑧 ∈ V
44		fveq2 6908	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → (card‘𝑥) = (card‘𝑧))
45		id 23	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → 𝑥 = 𝑧)
46	44, 45	eqeq12d 2779	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → ((card‘𝑥) = 𝑥 ↔ (card‘𝑧) = 𝑧))
47		sseq2 3939	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (ω ⊆ 𝑥 ↔ ω ⊆ 𝑧))
48		eqeq1 2767	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → (𝑥 = (ℵ‘𝑦) ↔ 𝑧 = (ℵ‘𝑦)))
49	48	notbid 321	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → (¬ 𝑥 = (ℵ‘𝑦) ↔ ¬ 𝑧 = (ℵ‘𝑦)))
50	49	ralbidv 3189	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (∀𝑦 ∈ 𝐴 ¬ 𝑥 = (ℵ‘𝑦) ↔ ∀𝑦 ∈ 𝐴 ¬ 𝑧 = (ℵ‘𝑦)))
51	46, 47, 50	3anbi123d 1465	. . . . . . 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 3636	. . . . . 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 1105	. . . . . 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 278	. . . . 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 323	. . . 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 255	. . 3 ⊢ (𝐴 ∈ On → (𝑧 ∈ (ℵ‘𝐴) → ¬ 𝑧 ∈ {𝑥 ∣ ((card‘𝑥) = 𝑥 ∧ ω ⊆ 𝑥 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑥 = (ℵ‘𝑦))}))
57	56	ralrimiv 3157	. 2 ⊢ (𝐴 ∈ On → ∀𝑧 ∈ (ℵ‘𝐴) ¬ 𝑧 ∈ {𝑥 ∣ ((card‘𝑥) = 𝑥 ∧ ω ⊆ 𝑥 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑥 = (ℵ‘𝑦))})
58		cardon 10052	. . . . . 6 ⊢ (card‘𝑥) ∈ On
59		eleq1 2852	. . . . . 6 ⊢ ((card‘𝑥) = 𝑥 → ((card‘𝑥) ∈ On ↔ 𝑥 ∈ On))
60	58, 59	mpbii 236	. . . . 5 ⊢ ((card‘𝑥) = 𝑥 → 𝑥 ∈ On)
61	60	3ad2ant1 1151	. . . 4 ⊢ (((card‘𝑥) = 𝑥 ∧ ω ⊆ 𝑥 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑥 = (ℵ‘𝑦)) → 𝑥 ∈ On)
62	61	abssi 3996	. . 3 ⊢ {𝑥 ∣ ((card‘𝑥) = 𝑥 ∧ ω ⊆ 𝑥 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑥 = (ℵ‘𝑦))} ⊆ On
63		oneqmini 6434	. . 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 596	1 ⊢ (𝐴 ∈ On → (ℵ‘𝐴) = ∩ {𝑥 ∣ ((card‘𝑥) = 𝑥 ∧ ω ⊆ 𝑥 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑥 = (ℵ‘𝑦))})

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147  {cab 2741  ∀wral 3080  ∃wrex 3090   ⊆ wss 3900  ∩ cint 4910  Oncon0 6380  ‘cfv 6558  ωcom 7901  cardccrd 10043  ℵcale 10044

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7775  ax-inf2 9669

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-se 5609  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6322  df-ord 6383  df-on 6384  df-lim 6385  df-suc 6386  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-isom 6567  df-riota 7400  df-ov 7446  df-om 7902  df-2nd 8026  df-frecs 8327  df-wrecs 8358  df-recs 8407  df-rdg 8446  df-1o 8504  df-er 8745  df-en 9002  df-dom 9003  df-sdom 9004  df-fin 9005  df-oi 9531  df-har 9578  df-card 10047  df-aleph 10048

This theorem is used by: (None)