Theorem alephval3 10052

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 10012	. . . 4 ⊢ (card‘(ℵ‘𝐴)) = (ℵ‘𝐴)
2	1	a1i 11	. . 3 ⊢ (𝐴 ∈ On → (card‘(ℵ‘𝐴)) = (ℵ‘𝐴))
3		alephgeom 10024	. . . 4 ⊢ (𝐴 ∈ On ↔ ω ⊆ (ℵ‘𝐴))
4	3	biimpi 218	. . 3 ⊢ (𝐴 ∈ On → ω ⊆ (ℵ‘𝐴))
5		alephord2i 10019	. . . . 5 ⊢ (𝐴 ∈ On → (𝑦 ∈ 𝐴 → (ℵ‘𝑦) ∈ (ℵ‘𝐴)))
6		alephon 10011	. . . . . . . 8 ⊢ (ℵ‘𝑦) ∈ On
7	6	onirri 6445	. . . . . . 7 ⊢ ¬ (ℵ‘𝑦) ∈ (ℵ‘𝑦)
8		eleq2 2841	. . . . . . 7 ⊢ ((ℵ‘𝐴) = (ℵ‘𝑦) → ((ℵ‘𝑦) ∈ (ℵ‘𝐴) ↔ (ℵ‘𝑦) ∈ (ℵ‘𝑦)))
9	7, 8	mtbiri 329	. . . . . 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 3143	. . 3 ⊢ (𝐴 ∈ On → ∀𝑦 ∈ 𝐴 ¬ (ℵ‘𝐴) = (ℵ‘𝑦))
13		fvex 6865	. . . 4 ⊢ (ℵ‘𝐴) ∈ V
14		fveq2 6852	. . . . . 6 ⊢ (𝑥 = (ℵ‘𝐴) → (card‘𝑥) = (card‘(ℵ‘𝐴)))
15		id 22	. . . . . 6 ⊢ (𝑥 = (ℵ‘𝐴) → 𝑥 = (ℵ‘𝐴))
16	14, 15	eqeq12d 2768	. . . . 5 ⊢ (𝑥 = (ℵ‘𝐴) → ((card‘𝑥) = 𝑥 ↔ (card‘(ℵ‘𝐴)) = (ℵ‘𝐴)))
17		sseq2 3953	. . . . 5 ⊢ (𝑥 = (ℵ‘𝐴) → (ω ⊆ 𝑥 ↔ ω ⊆ (ℵ‘𝐴)))
18		eqeq1 2756	. . . . . . 7 ⊢ (𝑥 = (ℵ‘𝐴) → (𝑥 = (ℵ‘𝑦) ↔ (ℵ‘𝐴) = (ℵ‘𝑦)))
19	18	notbid 320	. . . . . 6 ⊢ (𝑥 = (ℵ‘𝐴) → (¬ 𝑥 = (ℵ‘𝑦) ↔ ¬ (ℵ‘𝐴) = (ℵ‘𝑦)))
20	19	ralbidv 3175	. . . . 5 ⊢ (𝑥 = (ℵ‘𝐴) → (∀𝑦 ∈ 𝐴 ¬ 𝑥 = (ℵ‘𝑦) ↔ ∀𝑦 ∈ 𝐴 ¬ (ℵ‘𝐴) = (ℵ‘𝑦)))
21	16, 17, 20	3anbi123d 1447	. . . 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 3629	. . 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 1353	. 2 ⊢ (𝐴 ∈ On → (ℵ‘𝐴) ∈ {𝑥 ∣ ((card‘𝑥) = 𝑥 ∧ ω ⊆ 𝑥 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑥 = (ℵ‘𝑦))})
24		eleq1 2840	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = (ℵ‘𝑦) → (𝑧 ∈ (ℵ‘𝐴) ↔ (ℵ‘𝑦) ∈ (ℵ‘𝐴)))
25		alephord2 10018	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ On ∧ 𝐴 ∈ On) → (𝑦 ∈ 𝐴 ↔ (ℵ‘𝑦) ∈ (ℵ‘𝐴)))
26	25	bicomd 225	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ On ∧ 𝐴 ∈ On) → ((ℵ‘𝑦) ∈ (ℵ‘𝐴) ↔ 𝑦 ∈ 𝐴))
27	24, 26	sylan9bbr 517	. . . . . . . . . . . . . 14 ⊢ (((𝑦 ∈ On ∧ 𝐴 ∈ On) ∧ 𝑧 = (ℵ‘𝑦)) → (𝑧 ∈ (ℵ‘𝐴) ↔ 𝑦 ∈ 𝐴))
28	27	biimpcd 251	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ (ℵ‘𝐴) → (((𝑦 ∈ On ∧ 𝐴 ∈ On) ∧ 𝑧 = (ℵ‘𝑦)) → 𝑦 ∈ 𝐴))
29		simpr 487	. . . . . . . . . . . . 13 ⊢ (((𝑦 ∈ On ∧ 𝐴 ∈ On) ∧ 𝑧 = (ℵ‘𝑦)) → 𝑧 = (ℵ‘𝑦))
30	28, 29	jca2 520	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ (ℵ‘𝐴) → (((𝑦 ∈ On ∧ 𝐴 ∈ On) ∧ 𝑧 = (ℵ‘𝑦)) → (𝑦 ∈ 𝐴 ∧ 𝑧 = (ℵ‘𝑦))))
31	30	exp4c 435	. . . . . . . . . . 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 424	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ 𝑧 ∈ (ℵ‘𝐴)) → ((𝑦 ∈ On ∧ 𝑧 = (ℵ‘𝑦)) → (𝑦 ∈ 𝐴 ∧ 𝑧 = (ℵ‘𝑦))))
34	33	reximdv2 3162	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ 𝑧 ∈ (ℵ‘𝐴)) → (∃𝑦 ∈ On 𝑧 = (ℵ‘𝑦) → ∃𝑦 ∈ 𝐴 𝑧 = (ℵ‘𝑦)))
35		cardalephex 10032	. . . . . . . . 9 ⊢ (ω ⊆ 𝑧 → ((card‘𝑧) = 𝑧 ↔ ∃𝑦 ∈ On 𝑧 = (ℵ‘𝑦)))
36	35	biimpac 481	. . . . . . . 8 ⊢ (((card‘𝑧) = 𝑧 ∧ ω ⊆ 𝑧) → ∃𝑦 ∈ On 𝑧 = (ℵ‘𝑦))
37	34, 36	impel 512	. . . . . . 7 ⊢ (((𝐴 ∈ On ∧ 𝑧 ∈ (ℵ‘𝐴)) ∧ ((card‘𝑧) = 𝑧 ∧ ω ⊆ 𝑧)) → ∃𝑦 ∈ 𝐴 𝑧 = (ℵ‘𝑦))
38		dfrex2 3079	. . . . . . 7 ⊢ (∃𝑦 ∈ 𝐴 𝑧 = (ℵ‘𝑦) ↔ ¬ ∀𝑦 ∈ 𝐴 ¬ 𝑧 = (ℵ‘𝑦))
39	37, 38	sylib 220	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ 𝑧 ∈ (ℵ‘𝐴)) ∧ ((card‘𝑧) = 𝑧 ∧ ω ⊆ 𝑧)) → ¬ ∀𝑦 ∈ 𝐴 ¬ 𝑧 = (ℵ‘𝑦))
40		nan 838	. . . . . 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 233	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝑧 ∈ (ℵ‘𝐴)) → ¬ (((card‘𝑧) = 𝑧 ∧ ω ⊆ 𝑧) ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑧 = (ℵ‘𝑦)))
42	41	ex 415	. . . 4 ⊢ (𝐴 ∈ On → (𝑧 ∈ (ℵ‘𝐴) → ¬ (((card‘𝑧) = 𝑧 ∧ ω ⊆ 𝑧) ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑧 = (ℵ‘𝑦))))
43		vex 3448	. . . . . . 7 ⊢ 𝑧 ∈ V
44		fveq2 6852	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → (card‘𝑥) = (card‘𝑧))
45		id 22	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → 𝑥 = 𝑧)
46	44, 45	eqeq12d 2768	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → ((card‘𝑥) = 𝑥 ↔ (card‘𝑧) = 𝑧))
47		sseq2 3953	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (ω ⊆ 𝑥 ↔ ω ⊆ 𝑧))
48		eqeq1 2756	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → (𝑥 = (ℵ‘𝑦) ↔ 𝑧 = (ℵ‘𝑦)))
49	48	notbid 320	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → (¬ 𝑥 = (ℵ‘𝑦) ↔ ¬ 𝑧 = (ℵ‘𝑦)))
50	49	ralbidv 3175	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (∀𝑦 ∈ 𝐴 ¬ 𝑥 = (ℵ‘𝑦) ↔ ∀𝑦 ∈ 𝐴 ¬ 𝑧 = (ℵ‘𝑦)))
51	46, 47, 50	3anbi123d 1447	. . . . . . 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 3629	. . . . . 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 1097	. . . . . 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 277	. . . . 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 322	. . . 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 254	. . 3 ⊢ (𝐴 ∈ On → (𝑧 ∈ (ℵ‘𝐴) → ¬ 𝑧 ∈ {𝑥 ∣ ((card‘𝑥) = 𝑥 ∧ ω ⊆ 𝑥 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑥 = (ℵ‘𝑦))}))
57	56	ralrimiv 3143	. 2 ⊢ (𝐴 ∈ On → ∀𝑧 ∈ (ℵ‘𝐴) ¬ 𝑧 ∈ {𝑥 ∣ ((card‘𝑥) = 𝑥 ∧ ω ⊆ 𝑥 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑥 = (ℵ‘𝑦))})
58		cardon 9888	. . . . . 6 ⊢ (card‘𝑥) ∈ On
59		eleq1 2840	. . . . . 6 ⊢ ((card‘𝑥) = 𝑥 → ((card‘𝑥) ∈ On ↔ 𝑥 ∈ On))
60	58, 59	mpbii 235	. . . . 5 ⊢ ((card‘𝑥) = 𝑥 → 𝑥 ∈ On)
61	60	3ad2ant1 1142	. . . 4 ⊢ (((card‘𝑥) = 𝑥 ∧ ω ⊆ 𝑥 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑥 = (ℵ‘𝑦)) → 𝑥 ∈ On)
62	61	abssi 4012	. . 3 ⊢ {𝑥 ∣ ((card‘𝑥) = 𝑥 ∧ ω ⊆ 𝑥 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑥 = (ℵ‘𝑦))} ⊆ On
63		oneqmini 6384	. . 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 592	1 ⊢ (𝐴 ∈ On → (ℵ‘𝐴) = ∩ {𝑥 ∣ ((card‘𝑥) = 𝑥 ∧ ω ⊆ 𝑥 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑥 = (ℵ‘𝑦))})

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 398   ∧ w3a 1095   = wceq 1550   ∈ wcel 2132  {cab 2730  ∀wral 3066  ∃wrex 3076   ⊆ wss 3895  ∩ cint 4895  Oncon0 6331  ‘cfv 6506  ωcom 7831  cardccrd 9879  ℵcale 9880

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1805  ax-4 1819  ax-5 1920  ax-6 1977  ax-7 2018  ax-8 2134  ax-9 2142  ax-10 2165  ax-11 2181  ax-12 2202  ax-ext 2724  ax-rep 5217  ax-sep 5236  ax-nul 5246  ax-pow 5312  ax-pr 5380  ax-un 7703  ax-inf2 9582

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3or 1096  df-3an 1097  df-tru 1553  df-fal 1563  df-ex 1790  df-nf 1794  df-sb 2081  df-mo 2556  df-eu 2586  df-clab 2731  df-cleq 2744  df-clel 2827  df-nfc 2901  df-ne 2948  df-ral 3067  df-rex 3077  df-rmo 3357  df-reu 3358  df-rab 3405  df-v 3446  df-sbc 3736  df-csb 3844  df-dif 3898  df-un 3900  df-in 3902  df-ss 3912  df-pss 3915  df-nul 4277  df-if 4471  df-pw 4547  df-sn 4573  df-pr 4575  df-op 4579  df-uni 4856  df-int 4896  df-iun 4941  df-br 5091  df-opab 5153  df-mpt 5172  df-tr 5198  df-id 5531  df-eprel 5536  df-po 5544  df-so 5545  df-fr 5589  df-se 5590  df-we 5591  df-xp 5642  df-rel 5643  df-cnv 5644  df-co 5645  df-dm 5646  df-rn 5647  df-res 5648  df-ima 5649  df-pred 6273  df-ord 6334  df-on 6335  df-lim 6336  df-suc 6337  df-iota 6462  df-fun 6508  df-fn 6509  df-f 6510  df-f1 6511  df-fo 6512  df-f1o 6513  df-fv 6514  df-isom 6515  df-riota 7338  df-ov 7384  df-om 7832  df-2nd 7956  df-frecs 8246  df-wrecs 8277  df-recs 8326  df-rdg 8365  df-1o 8421  df-er 8662  df-en 8913  df-dom 8914  df-sdom 8915  df-fin 8916  df-oi 9444  df-har 9491  df-card 9883  df-aleph 9884

This theorem is referenced by: (None)