Theorem alephval3 10150

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 10110	. . . 4 ⊢ (card‘(ℵ‘𝐴)) = (ℵ‘𝐴)
2	1	a1i 11	. . 3 ⊢ (𝐴 ∈ On → (card‘(ℵ‘𝐴)) = (ℵ‘𝐴))
3		alephgeom 10122	. . . 4 ⊢ (𝐴 ∈ On ↔ ω ⊆ (ℵ‘𝐴))
4	3	biimpi 216	. . 3 ⊢ (𝐴 ∈ On → ω ⊆ (ℵ‘𝐴))
5		alephord2i 10117	. . . . 5 ⊢ (𝐴 ∈ On → (𝑦 ∈ 𝐴 → (ℵ‘𝑦) ∈ (ℵ‘𝐴)))
6		elirr 9637	. . . . . . 7 ⊢ ¬ (ℵ‘𝑦) ∈ (ℵ‘𝑦)
7		eleq2 2830	. . . . . . 7 ⊢ ((ℵ‘𝐴) = (ℵ‘𝑦) → ((ℵ‘𝑦) ∈ (ℵ‘𝐴) ↔ (ℵ‘𝑦) ∈ (ℵ‘𝑦)))
8	6, 7	mtbiri 327	. . . . . 6 ⊢ ((ℵ‘𝐴) = (ℵ‘𝑦) → ¬ (ℵ‘𝑦) ∈ (ℵ‘𝐴))
9	8	con2i 139	. . . . 5 ⊢ ((ℵ‘𝑦) ∈ (ℵ‘𝐴) → ¬ (ℵ‘𝐴) = (ℵ‘𝑦))
10	5, 9	syl6 35	. . . 4 ⊢ (𝐴 ∈ On → (𝑦 ∈ 𝐴 → ¬ (ℵ‘𝐴) = (ℵ‘𝑦)))
11	10	ralrimiv 3145	. . 3 ⊢ (𝐴 ∈ On → ∀𝑦 ∈ 𝐴 ¬ (ℵ‘𝐴) = (ℵ‘𝑦))
12		fvex 6919	. . . 4 ⊢ (ℵ‘𝐴) ∈ V
13		fveq2 6906	. . . . . 6 ⊢ (𝑥 = (ℵ‘𝐴) → (card‘𝑥) = (card‘(ℵ‘𝐴)))
14		id 22	. . . . . 6 ⊢ (𝑥 = (ℵ‘𝐴) → 𝑥 = (ℵ‘𝐴))
15	13, 14	eqeq12d 2753	. . . . 5 ⊢ (𝑥 = (ℵ‘𝐴) → ((card‘𝑥) = 𝑥 ↔ (card‘(ℵ‘𝐴)) = (ℵ‘𝐴)))
16		sseq2 4010	. . . . 5 ⊢ (𝑥 = (ℵ‘𝐴) → (ω ⊆ 𝑥 ↔ ω ⊆ (ℵ‘𝐴)))
17		eqeq1 2741	. . . . . . 7 ⊢ (𝑥 = (ℵ‘𝐴) → (𝑥 = (ℵ‘𝑦) ↔ (ℵ‘𝐴) = (ℵ‘𝑦)))
18	17	notbid 318	. . . . . 6 ⊢ (𝑥 = (ℵ‘𝐴) → (¬ 𝑥 = (ℵ‘𝑦) ↔ ¬ (ℵ‘𝐴) = (ℵ‘𝑦)))
19	18	ralbidv 3178	. . . . 5 ⊢ (𝑥 = (ℵ‘𝐴) → (∀𝑦 ∈ 𝐴 ¬ 𝑥 = (ℵ‘𝑦) ↔ ∀𝑦 ∈ 𝐴 ¬ (ℵ‘𝐴) = (ℵ‘𝑦)))
20	15, 16, 19	3anbi123d 1438	. . . 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))))
21	12, 20	elab 3679	. . 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)))
22	2, 4, 11, 21	syl3anbrc 1344	. 2 ⊢ (𝐴 ∈ On → (ℵ‘𝐴) ∈ {𝑥 ∣ ((card‘𝑥) = 𝑥 ∧ ω ⊆ 𝑥 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑥 = (ℵ‘𝑦))})
23		eleq1 2829	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = (ℵ‘𝑦) → (𝑧 ∈ (ℵ‘𝐴) ↔ (ℵ‘𝑦) ∈ (ℵ‘𝐴)))
24		alephord2 10116	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ On ∧ 𝐴 ∈ On) → (𝑦 ∈ 𝐴 ↔ (ℵ‘𝑦) ∈ (ℵ‘𝐴)))
25	24	bicomd 223	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ On ∧ 𝐴 ∈ On) → ((ℵ‘𝑦) ∈ (ℵ‘𝐴) ↔ 𝑦 ∈ 𝐴))
26	23, 25	sylan9bbr 510	. . . . . . . . . . . . . 14 ⊢ (((𝑦 ∈ On ∧ 𝐴 ∈ On) ∧ 𝑧 = (ℵ‘𝑦)) → (𝑧 ∈ (ℵ‘𝐴) ↔ 𝑦 ∈ 𝐴))
27	26	biimpcd 249	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ (ℵ‘𝐴) → (((𝑦 ∈ On ∧ 𝐴 ∈ On) ∧ 𝑧 = (ℵ‘𝑦)) → 𝑦 ∈ 𝐴))
28		simpr 484	. . . . . . . . . . . . 13 ⊢ (((𝑦 ∈ On ∧ 𝐴 ∈ On) ∧ 𝑧 = (ℵ‘𝑦)) → 𝑧 = (ℵ‘𝑦))
29	27, 28	jca2 513	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ (ℵ‘𝐴) → (((𝑦 ∈ On ∧ 𝐴 ∈ On) ∧ 𝑧 = (ℵ‘𝑦)) → (𝑦 ∈ 𝐴 ∧ 𝑧 = (ℵ‘𝑦))))
30	29	exp4c 432	. . . . . . . . . . 11 ⊢ (𝑧 ∈ (ℵ‘𝐴) → (𝑦 ∈ On → (𝐴 ∈ On → (𝑧 = (ℵ‘𝑦) → (𝑦 ∈ 𝐴 ∧ 𝑧 = (ℵ‘𝑦))))))
31	30	com3r 87	. . . . . . . . . 10 ⊢ (𝐴 ∈ On → (𝑧 ∈ (ℵ‘𝐴) → (𝑦 ∈ On → (𝑧 = (ℵ‘𝑦) → (𝑦 ∈ 𝐴 ∧ 𝑧 = (ℵ‘𝑦))))))
32	31	imp4b 421	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ 𝑧 ∈ (ℵ‘𝐴)) → ((𝑦 ∈ On ∧ 𝑧 = (ℵ‘𝑦)) → (𝑦 ∈ 𝐴 ∧ 𝑧 = (ℵ‘𝑦))))
33	32	reximdv2 3164	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ 𝑧 ∈ (ℵ‘𝐴)) → (∃𝑦 ∈ On 𝑧 = (ℵ‘𝑦) → ∃𝑦 ∈ 𝐴 𝑧 = (ℵ‘𝑦)))
34		cardalephex 10130	. . . . . . . . 9 ⊢ (ω ⊆ 𝑧 → ((card‘𝑧) = 𝑧 ↔ ∃𝑦 ∈ On 𝑧 = (ℵ‘𝑦)))
35	34	biimpac 478	. . . . . . . 8 ⊢ (((card‘𝑧) = 𝑧 ∧ ω ⊆ 𝑧) → ∃𝑦 ∈ On 𝑧 = (ℵ‘𝑦))
36	33, 35	impel 505	. . . . . . 7 ⊢ (((𝐴 ∈ On ∧ 𝑧 ∈ (ℵ‘𝐴)) ∧ ((card‘𝑧) = 𝑧 ∧ ω ⊆ 𝑧)) → ∃𝑦 ∈ 𝐴 𝑧 = (ℵ‘𝑦))
37		dfrex2 3073	. . . . . . 7 ⊢ (∃𝑦 ∈ 𝐴 𝑧 = (ℵ‘𝑦) ↔ ¬ ∀𝑦 ∈ 𝐴 ¬ 𝑧 = (ℵ‘𝑦))
38	36, 37	sylib 218	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ 𝑧 ∈ (ℵ‘𝐴)) ∧ ((card‘𝑧) = 𝑧 ∧ ω ⊆ 𝑧)) → ¬ ∀𝑦 ∈ 𝐴 ¬ 𝑧 = (ℵ‘𝑦))
39		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)))
40	38, 39	mpbir 231	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝑧 ∈ (ℵ‘𝐴)) → ¬ (((card‘𝑧) = 𝑧 ∧ ω ⊆ 𝑧) ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑧 = (ℵ‘𝑦)))
41	40	ex 412	. . . 4 ⊢ (𝐴 ∈ On → (𝑧 ∈ (ℵ‘𝐴) → ¬ (((card‘𝑧) = 𝑧 ∧ ω ⊆ 𝑧) ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑧 = (ℵ‘𝑦))))
42		vex 3484	. . . . . . 7 ⊢ 𝑧 ∈ V
43		fveq2 6906	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → (card‘𝑥) = (card‘𝑧))
44		id 22	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → 𝑥 = 𝑧)
45	43, 44	eqeq12d 2753	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → ((card‘𝑥) = 𝑥 ↔ (card‘𝑧) = 𝑧))
46		sseq2 4010	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (ω ⊆ 𝑥 ↔ ω ⊆ 𝑧))
47		eqeq1 2741	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → (𝑥 = (ℵ‘𝑦) ↔ 𝑧 = (ℵ‘𝑦)))
48	47	notbid 318	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → (¬ 𝑥 = (ℵ‘𝑦) ↔ ¬ 𝑧 = (ℵ‘𝑦)))
49	48	ralbidv 3178	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (∀𝑦 ∈ 𝐴 ¬ 𝑥 = (ℵ‘𝑦) ↔ ∀𝑦 ∈ 𝐴 ¬ 𝑧 = (ℵ‘𝑦)))
50	45, 46, 49	3anbi123d 1438	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (((card‘𝑥) = 𝑥 ∧ ω ⊆ 𝑥 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑥 = (ℵ‘𝑦)) ↔ ((card‘𝑧) = 𝑧 ∧ ω ⊆ 𝑧 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑧 = (ℵ‘𝑦))))
51	42, 50	elab 3679	. . . . . 6 ⊢ (𝑧 ∈ {𝑥 ∣ ((card‘𝑥) = 𝑥 ∧ ω ⊆ 𝑥 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑥 = (ℵ‘𝑦))} ↔ ((card‘𝑧) = 𝑧 ∧ ω ⊆ 𝑧 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑧 = (ℵ‘𝑦)))
52		df-3an 1089	. . . . . 6 ⊢ (((card‘𝑧) = 𝑧 ∧ ω ⊆ 𝑧 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑧 = (ℵ‘𝑦)) ↔ (((card‘𝑧) = 𝑧 ∧ ω ⊆ 𝑧) ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑧 = (ℵ‘𝑦)))
53	51, 52	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)))
54	53	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)))
55	41, 54	imbitrrdi 252	. . 3 ⊢ (𝐴 ∈ On → (𝑧 ∈ (ℵ‘𝐴) → ¬ 𝑧 ∈ {𝑥 ∣ ((card‘𝑥) = 𝑥 ∧ ω ⊆ 𝑥 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑥 = (ℵ‘𝑦))}))
56	55	ralrimiv 3145	. 2 ⊢ (𝐴 ∈ On → ∀𝑧 ∈ (ℵ‘𝐴) ¬ 𝑧 ∈ {𝑥 ∣ ((card‘𝑥) = 𝑥 ∧ ω ⊆ 𝑥 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑥 = (ℵ‘𝑦))})
57		cardon 9984	. . . . . 6 ⊢ (card‘𝑥) ∈ On
58		eleq1 2829	. . . . . 6 ⊢ ((card‘𝑥) = 𝑥 → ((card‘𝑥) ∈ On ↔ 𝑥 ∈ On))
59	57, 58	mpbii 233	. . . . 5 ⊢ ((card‘𝑥) = 𝑥 → 𝑥 ∈ On)
60	59	3ad2ant1 1134	. . . 4 ⊢ (((card‘𝑥) = 𝑥 ∧ ω ⊆ 𝑥 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑥 = (ℵ‘𝑦)) → 𝑥 ∈ On)
61	60	abssi 4070	. . 3 ⊢ {𝑥 ∣ ((card‘𝑥) = 𝑥 ∧ ω ⊆ 𝑥 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑥 = (ℵ‘𝑦))} ⊆ On
62		oneqmini 6436	. . 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))}))
63	61, 62	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))})
64	22, 56, 63	syl2anc 584	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 1540   ∈ wcel 2108  {cab 2714  ∀wral 3061  ∃wrex 3070   ⊆ wss 3951  ∩ cint 4946  Oncon0 6384  ‘cfv 6561  ωcom 7887  cardccrd 9975  ℵcale 9976

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-reg 9632  ax-inf2 9681

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-om 7888  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-oi 9550  df-har 9597  df-card 9979  df-aleph 9980

This theorem is referenced by: (None)