Theorem cardaleph 10101

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 9956	. . . . . . . . 9 ⊢ (card‘𝐴) ∈ On
2		eleq1 2822	. . . . . . . . 9 ⊢ ((card‘𝐴) = 𝐴 → ((card‘𝐴) ∈ On ↔ 𝐴 ∈ On))
3	1, 2	mpbii 233	. . . . . . . 8 ⊢ ((card‘𝐴) = 𝐴 → 𝐴 ∈ On)
4		alephle 10100	. . . . . . . . 9 ⊢ (𝐴 ∈ On → 𝐴 ⊆ (ℵ‘𝐴))
5		fveq2 6875	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐴 → (ℵ‘𝑥) = (ℵ‘𝐴))
6	5	sseq2d 3991	. . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → (𝐴 ⊆ (ℵ‘𝑥) ↔ 𝐴 ⊆ (ℵ‘𝐴)))
7	6	rspcev 3601	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ 𝐴 ⊆ (ℵ‘𝐴)) → ∃𝑥 ∈ On 𝐴 ⊆ (ℵ‘𝑥))
8	4, 7	mpdan 687	. . . . . . . 8 ⊢ (𝐴 ∈ On → ∃𝑥 ∈ On 𝐴 ⊆ (ℵ‘𝑥))
9		nfcv 2898	. . . . . . . . . 10 ⊢ Ⅎ𝑥𝐴
10		nfcv 2898	. . . . . . . . . . 11 ⊢ Ⅎ𝑥ℵ
11		nfrab1 3436	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥{𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}
12	11	nfint 4932	. . . . . . . . . . 11 ⊢ Ⅎ𝑥∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}
13	10, 12	nffv 6885	. . . . . . . . . 10 ⊢ Ⅎ𝑥(ℵ‘∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)})
14	9, 13	nfss 3951	. . . . . . . . 9 ⊢ Ⅎ𝑥 𝐴 ⊆ (ℵ‘∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)})
15		fveq2 6875	. . . . . . . . . 10 ⊢ (𝑥 = ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} → (ℵ‘𝑥) = (ℵ‘∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}))
16	15	sseq2d 3991	. . . . . . . . 9 ⊢ (𝑥 = ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} → (𝐴 ⊆ (ℵ‘𝑥) ↔ 𝐴 ⊆ (ℵ‘∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)})))
17	14, 16	onminsb 7786	. . . . . . . 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 6875	. . . . . . . . 9 ⊢ (∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} = ∅ → (ℵ‘∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) = (ℵ‘∅))
21		aleph0 10078	. . . . . . . . 9 ⊢ (ℵ‘∅) = ω
22	20, 21	eqtrdi 2786	. . . . . . . 8 ⊢ (∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} = ∅ → (ℵ‘∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) = ω)
23	22	sseq1d 3990	. . . . . . 7 ⊢ (∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} = ∅ → ((ℵ‘∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) ⊆ 𝐴 ↔ ω ⊆ 𝐴))
24	23	biimprd 248	. . . . . 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 3974	. . . . 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	imbitrrdi 252	. . . 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 458	. 2 ⊢ ((ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴) → (∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} = ∅ → 𝐴 = (ℵ‘∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)})))
30		fveq2 6875	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → (ℵ‘𝑥) = (ℵ‘𝑦))
31	30	sseq2d 3991	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝐴 ⊆ (ℵ‘𝑥) ↔ 𝐴 ⊆ (ℵ‘𝑦)))
32	31	onnminsb 7791	. . . . . . . . 9 ⊢ (𝑦 ∈ On → (𝑦 ∈ ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} → ¬ 𝐴 ⊆ (ℵ‘𝑦)))
33		vex 3463	. . . . . . . . . . 11 ⊢ 𝑦 ∈ V
34	33	sucid 6435	. . . . . . . . . 10 ⊢ 𝑦 ∈ suc 𝑦
35		eleq2 2823	. . . . . . . . . 10 ⊢ (∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} = suc 𝑦 → (𝑦 ∈ ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ↔ 𝑦 ∈ suc 𝑦))
36	34, 35	mpbiri 258	. . . . . . . . 9 ⊢ (∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} = suc 𝑦 → 𝑦 ∈ ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)})
37	32, 36	impel 505	. . . . . . . 8 ⊢ ((𝑦 ∈ On ∧ ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} = suc 𝑦) → ¬ 𝐴 ⊆ (ℵ‘𝑦))
38	37	adantl 481	. . . . . . 7 ⊢ (((card‘𝐴) = 𝐴 ∧ (𝑦 ∈ On ∧ ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} = suc 𝑦)) → ¬ 𝐴 ⊆ (ℵ‘𝑦))
39		fveq2 6875	. . . . . . . . . . 11 ⊢ (∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} = suc 𝑦 → (ℵ‘∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) = (ℵ‘suc 𝑦))
40		alephsuc 10080	. . . . . . . . . . 11 ⊢ (𝑦 ∈ On → (ℵ‘suc 𝑦) = (har‘(ℵ‘𝑦)))
41	39, 40	sylan9eqr 2792	. . . . . . . . . 10 ⊢ ((𝑦 ∈ On ∧ ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} = suc 𝑦) → (ℵ‘∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) = (har‘(ℵ‘𝑦)))
42	41	eleq2d 2820	. . . . . . . . 9 ⊢ ((𝑦 ∈ On ∧ ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} = suc 𝑦) → (𝐴 ∈ (ℵ‘∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) ↔ 𝐴 ∈ (har‘(ℵ‘𝑦))))
43	42	biimpd 229	. . . . . . . 8 ⊢ ((𝑦 ∈ On ∧ ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} = suc 𝑦) → (𝐴 ∈ (ℵ‘∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) → 𝐴 ∈ (har‘(ℵ‘𝑦))))
44		elharval 9573	. . . . . . . . . 10 ⊢ (𝐴 ∈ (har‘(ℵ‘𝑦)) ↔ (𝐴 ∈ On ∧ 𝐴 ≼ (ℵ‘𝑦)))
45	44	simprbi 496	. . . . . . . . 9 ⊢ (𝐴 ∈ (har‘(ℵ‘𝑦)) → 𝐴 ≼ (ℵ‘𝑦))
46		onenon 9961	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ On → 𝐴 ∈ dom card)
47	3, 46	syl 17	. . . . . . . . . . 11 ⊢ ((card‘𝐴) = 𝐴 → 𝐴 ∈ dom card)
48		alephon 10081	. . . . . . . . . . . 12 ⊢ (ℵ‘𝑦) ∈ On
49		onenon 9961	. . . . . . . . . . . 12 ⊢ ((ℵ‘𝑦) ∈ On → (ℵ‘𝑦) ∈ dom card)
50	48, 49	ax-mp 5	. . . . . . . . . . 11 ⊢ (ℵ‘𝑦) ∈ dom card
51		carddom2 9989	. . . . . . . . . . 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 3984	. . . . . . . . . . 11 ⊢ ((card‘𝐴) = 𝐴 → ((card‘𝐴) ⊆ (card‘(ℵ‘𝑦)) ↔ 𝐴 ⊆ (card‘(ℵ‘𝑦))))
54		alephcard 10082	. . . . . . . . . . . 12 ⊢ (card‘(ℵ‘𝑦)) = (ℵ‘𝑦)
55	54	sseq2i 3988	. . . . . . . . . . 11 ⊢ (𝐴 ⊆ (card‘(ℵ‘𝑦)) ↔ 𝐴 ⊆ (ℵ‘𝑦))
56	53, 55	bitrdi 287	. . . . . . . . . 10 ⊢ ((card‘𝐴) = 𝐴 → ((card‘𝐴) ⊆ (card‘(ℵ‘𝑦)) ↔ 𝐴 ⊆ (ℵ‘𝑦)))
57	52, 56	bitr3d 281	. . . . . . . . 9 ⊢ ((card‘𝐴) = 𝐴 → (𝐴 ≼ (ℵ‘𝑦) ↔ 𝐴 ⊆ (ℵ‘𝑦)))
58	45, 57	imbitrid 244	. . . . . . . 8 ⊢ ((card‘𝐴) = 𝐴 → (𝐴 ∈ (har‘(ℵ‘𝑦)) → 𝐴 ⊆ (ℵ‘𝑦)))
59	43, 58	sylan9r 508	. . . . . . 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 198	. . . . . 6 ⊢ (((card‘𝐴) = 𝐴 ∧ (𝑦 ∈ On ∧ ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} = suc 𝑦)) → ¬ 𝐴 ∈ (ℵ‘∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}))
61	60	rexlimdvaa 3142	. . . . 5 ⊢ ((card‘𝐴) = 𝐴 → (∃𝑦 ∈ On ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} = suc 𝑦 → ¬ 𝐴 ∈ (ℵ‘∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)})))
62		onintrab2 7789	. . . . . . . . . . . . . 14 ⊢ (∃𝑥 ∈ On 𝐴 ⊆ (ℵ‘𝑥) ↔ ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∈ On)
63	8, 62	sylib 218	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ On → ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∈ On)
64		onelon 6377	. . . . . . . . . . . . 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 490	. . . . . . . . . . . 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 6468	. . . . . . . . . . 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 3135	. . . . . . . . 9 ⊢ (𝐴 ∈ On → ¬ ∃𝑦 ∈ ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}𝐴 ∈ (ℵ‘𝑦))
71	70	adantr 480	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ Lim ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) → ¬ ∃𝑦 ∈ ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}𝐴 ∈ (ℵ‘𝑦))
72		alephlim 10079	. . . . . . . . . . 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 2820	. . . . . . . . 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 4971	. . . . . . . . 9 ⊢ (𝐴 ∈ ∪ 𝑦 ∈ ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} (ℵ‘𝑦) ↔ ∃𝑦 ∈ ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}𝐴 ∈ (ℵ‘𝑦))
76	74, 75	bitrdi 287	. . . . . . . 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 325	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ Lim ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) → ¬ 𝐴 ∈ (ℵ‘∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}))
78	77	ex 412	. . . . . 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 859	. . . 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 10081	. . . . . . 7 ⊢ (ℵ‘∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) ∈ On
83		onsseleq 6393	. . . . . . 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 691	. . . . . 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 232	. . . . 5 ⊢ (𝐴 ∈ On → (𝐴 ∈ (ℵ‘∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) ∨ 𝐴 = (ℵ‘∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)})))
86	85	ord 864	. . . 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 481	. 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 6362	. . . . 5 ⊢ (∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∈ On → Ord ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)})
90		ordzsl 7838	. . . . . 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 1089	. . . . . 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 275	. . . . 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 218	. . . 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 481	. 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 860	1 ⊢ ((ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴) → 𝐴 = (ℵ‘∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   ∨ w3o 1085   = wceq 1540   ∈ wcel 2108  ∃wrex 3060  {crab 3415   ⊆ wss 3926  ∅c0 4308  ∩ cint 4922  ∪ ciun 4967   class class class wbr 5119  dom cdm 5654  Ord word 6351  Oncon0 6352  Lim wlim 6353  suc csuc 6354  ‘cfv 6530  ωcom 7859   ≼ cdom 8955  harchar 9568  cardccrd 9947  ℵcale 9948

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 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7727  ax-inf2 9653

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rmo 3359  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-int 4923  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-se 5607  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6290  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6483  df-fun 6532  df-fn 6533  df-f 6534  df-f1 6535  df-fo 6536  df-f1o 6537  df-fv 6538  df-isom 6539  df-riota 7360  df-ov 7406  df-om 7860  df-2nd 7987  df-frecs 8278  df-wrecs 8309  df-recs 8383  df-rdg 8422  df-1o 8478  df-er 8717  df-en 8958  df-dom 8959  df-sdom 8960  df-fin 8961  df-oi 9522  df-har 9569  df-card 9951  df-aleph 9952

This theorem is referenced by:  cardalephex  10102  tskcard  10793  minregex  43505