Theorem alephval3 10107

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 10067	. . . 4 ⊢ (card‘(ℵ‘𝐴)) = (ℵ‘𝐴)
2	1	a1i 11	. . 3 ⊢ (𝐴 ∈ On → (card‘(ℵ‘𝐴)) = (ℵ‘𝐴))
3		alephgeom 10079	. . . 4 ⊢ (𝐴 ∈ On ↔ ω ⊆ (ℵ‘𝐴))
4	3	biimpi 215	. . 3 ⊢ (𝐴 ∈ On → ω ⊆ (ℵ‘𝐴))
5		alephord2i 10074	. . . . 5 ⊢ (𝐴 ∈ On → (𝑦 ∈ 𝐴 → (ℵ‘𝑦) ∈ (ℵ‘𝐴)))
6		elirr 9594	. . . . . . 7 ⊢ ¬ (ℵ‘𝑦) ∈ (ℵ‘𝑦)
7		eleq2 2822	. . . . . . 7 ⊢ ((ℵ‘𝐴) = (ℵ‘𝑦) → ((ℵ‘𝑦) ∈ (ℵ‘𝐴) ↔ (ℵ‘𝑦) ∈ (ℵ‘𝑦)))
8	6, 7	mtbiri 326	. . . . . 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 6904	. . . 4 ⊢ (ℵ‘𝐴) ∈ V
13		fveq2 6891	. . . . . 6 ⊢ (𝑥 = (ℵ‘𝐴) → (card‘𝑥) = (card‘(ℵ‘𝐴)))
14		id 22	. . . . . 6 ⊢ (𝑥 = (ℵ‘𝐴) → 𝑥 = (ℵ‘𝐴))
15	13, 14	eqeq12d 2748	. . . . 5 ⊢ (𝑥 = (ℵ‘𝐴) → ((card‘𝑥) = 𝑥 ↔ (card‘(ℵ‘𝐴)) = (ℵ‘𝐴)))
16		sseq2 4008	. . . . 5 ⊢ (𝑥 = (ℵ‘𝐴) → (ω ⊆ 𝑥 ↔ ω ⊆ (ℵ‘𝐴)))
17		eqeq1 2736	. . . . . . 7 ⊢ (𝑥 = (ℵ‘𝐴) → (𝑥 = (ℵ‘𝑦) ↔ (ℵ‘𝐴) = (ℵ‘𝑦)))
18	17	notbid 317	. . . . . 6 ⊢ (𝑥 = (ℵ‘𝐴) → (¬ 𝑥 = (ℵ‘𝑦) ↔ ¬ (ℵ‘𝐴) = (ℵ‘𝑦)))
19	18	ralbidv 3177	. . . . 5 ⊢ (𝑥 = (ℵ‘𝐴) → (∀𝑦 ∈ 𝐴 ¬ 𝑥 = (ℵ‘𝑦) ↔ ∀𝑦 ∈ 𝐴 ¬ (ℵ‘𝐴) = (ℵ‘𝑦)))
20	15, 16, 19	3anbi123d 1436	. . . 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 3668	. . 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 1343	. 2 ⊢ (𝐴 ∈ On → (ℵ‘𝐴) ∈ {𝑥 ∣ ((card‘𝑥) = 𝑥 ∧ ω ⊆ 𝑥 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑥 = (ℵ‘𝑦))})
23		eleq1 2821	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = (ℵ‘𝑦) → (𝑧 ∈ (ℵ‘𝐴) ↔ (ℵ‘𝑦) ∈ (ℵ‘𝐴)))
24		alephord2 10073	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ On ∧ 𝐴 ∈ On) → (𝑦 ∈ 𝐴 ↔ (ℵ‘𝑦) ∈ (ℵ‘𝐴)))
25	24	bicomd 222	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ On ∧ 𝐴 ∈ On) → ((ℵ‘𝑦) ∈ (ℵ‘𝐴) ↔ 𝑦 ∈ 𝐴))
26	23, 25	sylan9bbr 511	. . . . . . . . . . . . . 14 ⊢ (((𝑦 ∈ On ∧ 𝐴 ∈ On) ∧ 𝑧 = (ℵ‘𝑦)) → (𝑧 ∈ (ℵ‘𝐴) ↔ 𝑦 ∈ 𝐴))
27	26	biimpcd 248	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ (ℵ‘𝐴) → (((𝑦 ∈ On ∧ 𝐴 ∈ On) ∧ 𝑧 = (ℵ‘𝑦)) → 𝑦 ∈ 𝐴))
28		simpr 485	. . . . . . . . . . . . 13 ⊢ (((𝑦 ∈ On ∧ 𝐴 ∈ On) ∧ 𝑧 = (ℵ‘𝑦)) → 𝑧 = (ℵ‘𝑦))
29	27, 28	jca2 514	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ (ℵ‘𝐴) → (((𝑦 ∈ On ∧ 𝐴 ∈ On) ∧ 𝑧 = (ℵ‘𝑦)) → (𝑦 ∈ 𝐴 ∧ 𝑧 = (ℵ‘𝑦))))
30	29	exp4c 433	. . . . . . . . . . 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 422	. . . . . . . . 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 10087	. . . . . . . . 9 ⊢ (ω ⊆ 𝑧 → ((card‘𝑧) = 𝑧 ↔ ∃𝑦 ∈ On 𝑧 = (ℵ‘𝑦)))
35	34	biimpac 479	. . . . . . . 8 ⊢ (((card‘𝑧) = 𝑧 ∧ ω ⊆ 𝑧) → ∃𝑦 ∈ On 𝑧 = (ℵ‘𝑦))
36	33, 35	impel 506	. . . . . . 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 217	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ 𝑧 ∈ (ℵ‘𝐴)) ∧ ((card‘𝑧) = 𝑧 ∧ ω ⊆ 𝑧)) → ¬ ∀𝑦 ∈ 𝐴 ¬ 𝑧 = (ℵ‘𝑦))
39		nan 828	. . . . . 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 230	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝑧 ∈ (ℵ‘𝐴)) → ¬ (((card‘𝑧) = 𝑧 ∧ ω ⊆ 𝑧) ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑧 = (ℵ‘𝑦)))
41	40	ex 413	. . . 4 ⊢ (𝐴 ∈ On → (𝑧 ∈ (ℵ‘𝐴) → ¬ (((card‘𝑧) = 𝑧 ∧ ω ⊆ 𝑧) ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑧 = (ℵ‘𝑦))))
42		vex 3478	. . . . . . 7 ⊢ 𝑧 ∈ V
43		fveq2 6891	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → (card‘𝑥) = (card‘𝑧))
44		id 22	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → 𝑥 = 𝑧)
45	43, 44	eqeq12d 2748	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → ((card‘𝑥) = 𝑥 ↔ (card‘𝑧) = 𝑧))
46		sseq2 4008	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (ω ⊆ 𝑥 ↔ ω ⊆ 𝑧))
47		eqeq1 2736	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → (𝑥 = (ℵ‘𝑦) ↔ 𝑧 = (ℵ‘𝑦)))
48	47	notbid 317	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → (¬ 𝑥 = (ℵ‘𝑦) ↔ ¬ 𝑧 = (ℵ‘𝑦)))
49	48	ralbidv 3177	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (∀𝑦 ∈ 𝐴 ¬ 𝑥 = (ℵ‘𝑦) ↔ ∀𝑦 ∈ 𝐴 ¬ 𝑧 = (ℵ‘𝑦)))
50	45, 46, 49	3anbi123d 1436	. . . . . . 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 3668	. . . . . 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 274	. . . . 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 319	. . . 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 251	. . 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 9941	. . . . . 6 ⊢ (card‘𝑥) ∈ On
58		eleq1 2821	. . . . . 6 ⊢ ((card‘𝑥) = 𝑥 → ((card‘𝑥) ∈ On ↔ 𝑥 ∈ On))
59	57, 58	mpbii 232	. . . . 5 ⊢ ((card‘𝑥) = 𝑥 → 𝑥 ∈ On)
60	59	3ad2ant1 1133	. . . 4 ⊢ (((card‘𝑥) = 𝑥 ∧ ω ⊆ 𝑥 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑥 = (ℵ‘𝑦)) → 𝑥 ∈ On)
61	60	abssi 4067	. . 3 ⊢ {𝑥 ∣ ((card‘𝑥) = 𝑥 ∧ ω ⊆ 𝑥 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑥 = (ℵ‘𝑦))} ⊆ On
62		oneqmini 6416	. . 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 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  {cab 2709  ∀wral 3061  ∃wrex 3070   ⊆ wss 3948  ∩ cint 4950  Oncon0 6364  ‘cfv 6543  ωcom 7857  cardccrd 9932  ℵcale 9933

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727  ax-reg 9589  ax-inf2 9638

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-se 5632  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-isom 6552  df-riota 7367  df-ov 7414  df-om 7858  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-1o 8468  df-er 8705  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-oi 9507  df-har 9554  df-card 9936  df-aleph 9937

This theorem is referenced by: (None)