Theorem cardaleph 10025

Description: Given any transfinite cardinal number 𝐴, there is exactly one aleph that is equal to it. Here we compute that aleph explicitly. (Contributed by NM, 9-Nov-2003.) (Revised by Mario Carneiro, 2-Feb-2013.)

Assertion

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

Distinct variable group:   𝑥,𝐴

Proof of Theorem cardaleph

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cardon 9880	. . . . . . . . 9 ⊢ (card‘𝐴) ∈ On
2		eleq1 2825	. . . . . . . . 9 ⊢ ((card‘𝐴) = 𝐴 → ((card‘𝐴) ∈ On ↔ 𝐴 ∈ On))
3	1, 2	mpbii 232	. . . . . . . 8 ⊢ ((card‘𝐴) = 𝐴 → 𝐴 ∈ On)
4		alephle 10024	. . . . . . . . 9 ⊢ (𝐴 ∈ On → 𝐴 ⊆ (ℵ‘𝐴))
5		fveq2 6842	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐴 → (ℵ‘𝑥) = (ℵ‘𝐴))
6	5	sseq2d 3976	. . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → (𝐴 ⊆ (ℵ‘𝑥) ↔ 𝐴 ⊆ (ℵ‘𝐴)))
7	6	rspcev 3581	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ 𝐴 ⊆ (ℵ‘𝐴)) → ∃𝑥 ∈ On 𝐴 ⊆ (ℵ‘𝑥))
8	4, 7	mpdan 685	. . . . . . . 8 ⊢ (𝐴 ∈ On → ∃𝑥 ∈ On 𝐴 ⊆ (ℵ‘𝑥))
9		nfcv 2907	. . . . . . . . . 10 ⊢ Ⅎ𝑥𝐴
10		nfcv 2907	. . . . . . . . . . 11 ⊢ Ⅎ𝑥ℵ
11		nfrab1 3426	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥{𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}
12	11	nfint 4917	. . . . . . . . . . 11 ⊢ Ⅎ𝑥∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}
13	10, 12	nffv 6852	. . . . . . . . . 10 ⊢ Ⅎ𝑥(ℵ‘∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)})
14	9, 13	nfss 3936	. . . . . . . . 9 ⊢ Ⅎ𝑥 𝐴 ⊆ (ℵ‘∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)})
15		fveq2 6842	. . . . . . . . . 10 ⊢ (𝑥 = ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} → (ℵ‘𝑥) = (ℵ‘∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}))
16	15	sseq2d 3976	. . . . . . . . 9 ⊢ (𝑥 = ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} → (𝐴 ⊆ (ℵ‘𝑥) ↔ 𝐴 ⊆ (ℵ‘∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)})))
17	14, 16	onminsb 7729	. . . . . . . 8 ⊢ (∃𝑥 ∈ On 𝐴 ⊆ (ℵ‘𝑥) → 𝐴 ⊆ (ℵ‘∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}))
18	3, 8, 17	3syl 18	. . . . . . 7 ⊢ ((card‘𝐴) = 𝐴 → 𝐴 ⊆ (ℵ‘∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}))
19	18	a1i 11	. . . . . 6 ⊢ (∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} = ∅ → ((card‘𝐴) = 𝐴 → 𝐴 ⊆ (ℵ‘∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)})))
20		fveq2 6842	. . . . . . . . 9 ⊢ (∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} = ∅ → (ℵ‘∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) = (ℵ‘∅))
21		aleph0 10002	. . . . . . . . 9 ⊢ (ℵ‘∅) = ω
22	20, 21	eqtrdi 2792	. . . . . . . 8 ⊢ (∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} = ∅ → (ℵ‘∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) = ω)
23	22	sseq1d 3975	. . . . . . 7 ⊢ (∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} = ∅ → ((ℵ‘∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) ⊆ 𝐴 ↔ ω ⊆ 𝐴))
24	23	biimprd 247	. . . . . 6 ⊢ (∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} = ∅ → (ω ⊆ 𝐴 → (ℵ‘∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) ⊆ 𝐴))
25	19, 24	anim12d 609	. . . . 5 ⊢ (∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} = ∅ → (((card‘𝐴) = 𝐴 ∧ ω ⊆ 𝐴) → (𝐴 ⊆ (ℵ‘∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) ∧ (ℵ‘∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) ⊆ 𝐴)))
26		eqss 3959	. . . . 5 ⊢ (𝐴 = (ℵ‘∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) ↔ (𝐴 ⊆ (ℵ‘∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) ∧ (ℵ‘∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) ⊆ 𝐴))
27	25, 26	syl6ibr 251	. . . 4 ⊢ (∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} = ∅ → (((card‘𝐴) = 𝐴 ∧ ω ⊆ 𝐴) → 𝐴 = (ℵ‘∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)})))
28	27	com12 32	. . 3 ⊢ (((card‘𝐴) = 𝐴 ∧ ω ⊆ 𝐴) → (∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} = ∅ → 𝐴 = (ℵ‘∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)})))
29	28	ancoms 459	. 2 ⊢ ((ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴) → (∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} = ∅ → 𝐴 = (ℵ‘∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)})))
30		fveq2 6842	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → (ℵ‘𝑥) = (ℵ‘𝑦))
31	30	sseq2d 3976	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝐴 ⊆ (ℵ‘𝑥) ↔ 𝐴 ⊆ (ℵ‘𝑦)))
32	31	onnminsb 7734	. . . . . . . . 9 ⊢ (𝑦 ∈ On → (𝑦 ∈ ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} → ¬ 𝐴 ⊆ (ℵ‘𝑦)))
33		vex 3449	. . . . . . . . . . 11 ⊢ 𝑦 ∈ V
34	33	sucid 6399	. . . . . . . . . 10 ⊢ 𝑦 ∈ suc 𝑦
35		eleq2 2826	. . . . . . . . . 10 ⊢ (∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} = suc 𝑦 → (𝑦 ∈ ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ↔ 𝑦 ∈ suc 𝑦))
36	34, 35	mpbiri 257	. . . . . . . . 9 ⊢ (∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} = suc 𝑦 → 𝑦 ∈ ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)})
37	32, 36	impel 506	. . . . . . . 8 ⊢ ((𝑦 ∈ On ∧ ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} = suc 𝑦) → ¬ 𝐴 ⊆ (ℵ‘𝑦))
38	37	adantl 482	. . . . . . 7 ⊢ (((card‘𝐴) = 𝐴 ∧ (𝑦 ∈ On ∧ ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} = suc 𝑦)) → ¬ 𝐴 ⊆ (ℵ‘𝑦))
39		fveq2 6842	. . . . . . . . . . 11 ⊢ (∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} = suc 𝑦 → (ℵ‘∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) = (ℵ‘suc 𝑦))
40		alephsuc 10004	. . . . . . . . . . 11 ⊢ (𝑦 ∈ On → (ℵ‘suc 𝑦) = (har‘(ℵ‘𝑦)))
41	39, 40	sylan9eqr 2798	. . . . . . . . . 10 ⊢ ((𝑦 ∈ On ∧ ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} = suc 𝑦) → (ℵ‘∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) = (har‘(ℵ‘𝑦)))
42	41	eleq2d 2823	. . . . . . . . 9 ⊢ ((𝑦 ∈ On ∧ ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} = suc 𝑦) → (𝐴 ∈ (ℵ‘∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) ↔ 𝐴 ∈ (har‘(ℵ‘𝑦))))
43	42	biimpd 228	. . . . . . . 8 ⊢ ((𝑦 ∈ On ∧ ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} = suc 𝑦) → (𝐴 ∈ (ℵ‘∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) → 𝐴 ∈ (har‘(ℵ‘𝑦))))
44		elharval 9497	. . . . . . . . . 10 ⊢ (𝐴 ∈ (har‘(ℵ‘𝑦)) ↔ (𝐴 ∈ On ∧ 𝐴 ≼ (ℵ‘𝑦)))
45	44	simprbi 497	. . . . . . . . 9 ⊢ (𝐴 ∈ (har‘(ℵ‘𝑦)) → 𝐴 ≼ (ℵ‘𝑦))
46		onenon 9885	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ On → 𝐴 ∈ dom card)
47	3, 46	syl 17	. . . . . . . . . . 11 ⊢ ((card‘𝐴) = 𝐴 → 𝐴 ∈ dom card)
48		alephon 10005	. . . . . . . . . . . 12 ⊢ (ℵ‘𝑦) ∈ On
49		onenon 9885	. . . . . . . . . . . 12 ⊢ ((ℵ‘𝑦) ∈ On → (ℵ‘𝑦) ∈ dom card)
50	48, 49	ax-mp 5	. . . . . . . . . . 11 ⊢ (ℵ‘𝑦) ∈ dom card
51		carddom2 9913	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ dom card ∧ (ℵ‘𝑦) ∈ dom card) → ((card‘𝐴) ⊆ (card‘(ℵ‘𝑦)) ↔ 𝐴 ≼ (ℵ‘𝑦)))
52	47, 50, 51	sylancl 586	. . . . . . . . . 10 ⊢ ((card‘𝐴) = 𝐴 → ((card‘𝐴) ⊆ (card‘(ℵ‘𝑦)) ↔ 𝐴 ≼ (ℵ‘𝑦)))
53		sseq1 3969	. . . . . . . . . . 11 ⊢ ((card‘𝐴) = 𝐴 → ((card‘𝐴) ⊆ (card‘(ℵ‘𝑦)) ↔ 𝐴 ⊆ (card‘(ℵ‘𝑦))))
54		alephcard 10006	. . . . . . . . . . . 12 ⊢ (card‘(ℵ‘𝑦)) = (ℵ‘𝑦)
55	54	sseq2i 3973	. . . . . . . . . . 11 ⊢ (𝐴 ⊆ (card‘(ℵ‘𝑦)) ↔ 𝐴 ⊆ (ℵ‘𝑦))
56	53, 55	bitrdi 286	. . . . . . . . . 10 ⊢ ((card‘𝐴) = 𝐴 → ((card‘𝐴) ⊆ (card‘(ℵ‘𝑦)) ↔ 𝐴 ⊆ (ℵ‘𝑦)))
57	52, 56	bitr3d 280	. . . . . . . . 9 ⊢ ((card‘𝐴) = 𝐴 → (𝐴 ≼ (ℵ‘𝑦) ↔ 𝐴 ⊆ (ℵ‘𝑦)))
58	45, 57	imbitrid 243	. . . . . . . 8 ⊢ ((card‘𝐴) = 𝐴 → (𝐴 ∈ (har‘(ℵ‘𝑦)) → 𝐴 ⊆ (ℵ‘𝑦)))
59	43, 58	sylan9r 509	. . . . . . 7 ⊢ (((card‘𝐴) = 𝐴 ∧ (𝑦 ∈ On ∧ ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} = suc 𝑦)) → (𝐴 ∈ (ℵ‘∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) → 𝐴 ⊆ (ℵ‘𝑦)))
60	38, 59	mtod 197	. . . . . 6 ⊢ (((card‘𝐴) = 𝐴 ∧ (𝑦 ∈ On ∧ ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} = suc 𝑦)) → ¬ 𝐴 ∈ (ℵ‘∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}))
61	60	rexlimdvaa 3153	. . . . 5 ⊢ ((card‘𝐴) = 𝐴 → (∃𝑦 ∈ On ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} = suc 𝑦 → ¬ 𝐴 ∈ (ℵ‘∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)})))
62		onintrab2 7732	. . . . . . . . . . . . . 14 ⊢ (∃𝑥 ∈ On 𝐴 ⊆ (ℵ‘𝑥) ↔ ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∈ On)
63	8, 62	sylib 217	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ On → ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∈ On)
64		onelon 6342	. . . . . . . . . . . . 13 ⊢ ((∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∈ On ∧ 𝑦 ∈ ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) → 𝑦 ∈ On)
65	63, 64	sylan 580	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) → 𝑦 ∈ On)
66	32	adantld 491	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ On → ((𝐴 ∈ On ∧ 𝑦 ∈ ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) → ¬ 𝐴 ⊆ (ℵ‘𝑦)))
67	65, 66	mpcom 38	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) → ¬ 𝐴 ⊆ (ℵ‘𝑦))
68	48	onelssi 6432	. . . . . . . . . . 11 ⊢ (𝐴 ∈ (ℵ‘𝑦) → 𝐴 ⊆ (ℵ‘𝑦))
69	67, 68	nsyl 140	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) → ¬ 𝐴 ∈ (ℵ‘𝑦))
70	69	nrexdv 3146	. . . . . . . . 9 ⊢ (𝐴 ∈ On → ¬ ∃𝑦 ∈ ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}𝐴 ∈ (ℵ‘𝑦))
71	70	adantr 481	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ Lim ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) → ¬ ∃𝑦 ∈ ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}𝐴 ∈ (ℵ‘𝑦))
72		alephlim 10003	. . . . . . . . . . 11 ⊢ ((∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∈ On ∧ Lim ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) → (ℵ‘∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) = ∪ 𝑦 ∈ ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} (ℵ‘𝑦))
73	63, 72	sylan 580	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ Lim ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) → (ℵ‘∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) = ∪ 𝑦 ∈ ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} (ℵ‘𝑦))
74	73	eleq2d 2823	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ Lim ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) → (𝐴 ∈ (ℵ‘∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) ↔ 𝐴 ∈ ∪ 𝑦 ∈ ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} (ℵ‘𝑦)))
75		eliun 4958	. . . . . . . . 9 ⊢ (𝐴 ∈ ∪ 𝑦 ∈ ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} (ℵ‘𝑦) ↔ ∃𝑦 ∈ ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}𝐴 ∈ (ℵ‘𝑦))
76	74, 75	bitrdi 286	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ Lim ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) → (𝐴 ∈ (ℵ‘∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) ↔ ∃𝑦 ∈ ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}𝐴 ∈ (ℵ‘𝑦)))
77	71, 76	mtbird 324	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ Lim ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) → ¬ 𝐴 ∈ (ℵ‘∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}))
78	77	ex 413	. . . . . 6 ⊢ (𝐴 ∈ On → (Lim ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} → ¬ 𝐴 ∈ (ℵ‘∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)})))
79	3, 78	syl 17	. . . . 5 ⊢ ((card‘𝐴) = 𝐴 → (Lim ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} → ¬ 𝐴 ∈ (ℵ‘∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)})))
80	61, 79	jaod 857	. . . 4 ⊢ ((card‘𝐴) = 𝐴 → ((∃𝑦 ∈ On ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} = suc 𝑦 ∨ Lim ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) → ¬ 𝐴 ∈ (ℵ‘∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)})))
81	8, 17	syl 17	. . . . . 6 ⊢ (𝐴 ∈ On → 𝐴 ⊆ (ℵ‘∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}))
82		alephon 10005	. . . . . . 7 ⊢ (ℵ‘∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) ∈ On
83		onsseleq 6358	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ (ℵ‘∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) ∈ On) → (𝐴 ⊆ (ℵ‘∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) ↔ (𝐴 ∈ (ℵ‘∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) ∨ 𝐴 = (ℵ‘∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}))))
84	82, 83	mpan2 689	. . . . . 6 ⊢ (𝐴 ∈ On → (𝐴 ⊆ (ℵ‘∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) ↔ (𝐴 ∈ (ℵ‘∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) ∨ 𝐴 = (ℵ‘∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}))))
85	81, 84	mpbid 231	. . . . 5 ⊢ (𝐴 ∈ On → (𝐴 ∈ (ℵ‘∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) ∨ 𝐴 = (ℵ‘∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)})))
86	85	ord 862	. . . 4 ⊢ (𝐴 ∈ On → (¬ 𝐴 ∈ (ℵ‘∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) → 𝐴 = (ℵ‘∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)})))
87	3, 80, 86	sylsyld 61	. . 3 ⊢ ((card‘𝐴) = 𝐴 → ((∃𝑦 ∈ On ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} = suc 𝑦 ∨ Lim ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) → 𝐴 = (ℵ‘∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)})))
88	87	adantl 482	. 2 ⊢ ((ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴) → ((∃𝑦 ∈ On ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} = suc 𝑦 ∨ Lim ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) → 𝐴 = (ℵ‘∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)})))
89		eloni 6327	. . . . 5 ⊢ (∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∈ On → Ord ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)})
90		ordzsl 7781	. . . . . 6 ⊢ (Ord ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ↔ (∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} = ∅ ∨ ∃𝑦 ∈ On ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} = suc 𝑦 ∨ Lim ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}))
91		3orass 1090	. . . . . 6 ⊢ ((∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} = ∅ ∨ ∃𝑦 ∈ On ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} = suc 𝑦 ∨ Lim ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) ↔ (∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} = ∅ ∨ (∃𝑦 ∈ On ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} = suc 𝑦 ∨ Lim ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)})))
92	90, 91	bitri 274	. . . . 5 ⊢ (Ord ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ↔ (∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} = ∅ ∨ (∃𝑦 ∈ On ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} = suc 𝑦 ∨ Lim ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)})))
93	89, 92	sylib 217	. . . 4 ⊢ (∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∈ On → (∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} = ∅ ∨ (∃𝑦 ∈ On ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} = suc 𝑦 ∨ Lim ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)})))
94	3, 63, 93	3syl 18	. . 3 ⊢ ((card‘𝐴) = 𝐴 → (∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} = ∅ ∨ (∃𝑦 ∈ On ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} = suc 𝑦 ∨ Lim ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)})))
95	94	adantl 482	. 2 ⊢ ((ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴) → (∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} = ∅ ∨ (∃𝑦 ∈ On ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} = suc 𝑦 ∨ Lim ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)})))
96	29, 88, 95	mpjaod 858	1 ⊢ ((ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴) → 𝐴 = (ℵ‘∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   ∨ wo 845   ∨ w3o 1086   = wceq 1541   ∈ wcel 2106  ∃wrex 3073  {crab 3407   ⊆ wss 3910  ∅c0 4282  ∩ cint 4907  ∪ ciun 4954   class class class wbr 5105  dom cdm 5633  Ord word 6316  Oncon0 6317  Lim wlim 6318  suc csuc 6319  ‘cfv 6496  ωcom 7802   ≼ cdom 8881  harchar 9492  cardccrd 9871  ℵcale 9872

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 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-inf2 9577

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 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-om 7803  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-oi 9446  df-har 9493  df-card 9875  df-aleph 9876

This theorem is referenced by:  cardalephex  10026  tskcard  10717  minregex  41796