Theorem minregex 43983

Description: Given any cardinal number 𝐴, there exists an argument 𝑥, which yields the least regular uncountable value of ℵ which is greater to or equal to 𝐴. This proof uses AC. (Contributed by RP, 23-Nov-2023.)

Assertion

Ref	Expression
minregex	⊢ (𝐴 ∈ (ran card ∖ ω) → ∃𝑥 ∈ On 𝑥 = ∩ {𝑦 ∈ On ∣ (∅ ∈ 𝑦 ∧ 𝐴 ⊆ (ℵ‘𝑦) ∧ (cf‘(ℵ‘𝑦)) = (ℵ‘𝑦))})

Distinct variable group:   𝑥,𝐴,𝑦

Proof of Theorem minregex

Step	Hyp	Ref	Expression
1		eldif 3900	. . . . . . 7 ⊢ (𝐴 ∈ (ran card ∖ ω) ↔ (𝐴 ∈ ran card ∧ ¬ 𝐴 ∈ ω))
2		omelon 9562	. . . . . . . . . 10 ⊢ ω ∈ On
3		cardon 9863	. . . . . . . . . . 11 ⊢ (card‘𝐴) ∈ On
4		eleq1 2825	. . . . . . . . . . 11 ⊢ ((card‘𝐴) = 𝐴 → ((card‘𝐴) ∈ On ↔ 𝐴 ∈ On))
5	3, 4	mpbii 233	. . . . . . . . . 10 ⊢ ((card‘𝐴) = 𝐴 → 𝐴 ∈ On)
6		ontri1 6353	. . . . . . . . . 10 ⊢ ((ω ∈ On ∧ 𝐴 ∈ On) → (ω ⊆ 𝐴 ↔ ¬ 𝐴 ∈ ω))
7	2, 5, 6	sylancr 588	. . . . . . . . 9 ⊢ ((card‘𝐴) = 𝐴 → (ω ⊆ 𝐴 ↔ ¬ 𝐴 ∈ ω))
8	7	pm5.32i 574	. . . . . . . 8 ⊢ (((card‘𝐴) = 𝐴 ∧ ω ⊆ 𝐴) ↔ ((card‘𝐴) = 𝐴 ∧ ¬ 𝐴 ∈ ω))
9		iscard4 43982	. . . . . . . . 9 ⊢ ((card‘𝐴) = 𝐴 ↔ 𝐴 ∈ ran card)
10	9	anbi1i 625	. . . . . . . 8 ⊢ (((card‘𝐴) = 𝐴 ∧ ¬ 𝐴 ∈ ω) ↔ (𝐴 ∈ ran card ∧ ¬ 𝐴 ∈ ω))
11	8, 10	bitr2i 276	. . . . . . 7 ⊢ ((𝐴 ∈ ran card ∧ ¬ 𝐴 ∈ ω) ↔ ((card‘𝐴) = 𝐴 ∧ ω ⊆ 𝐴))
12		ancom 460	. . . . . . 7 ⊢ (((card‘𝐴) = 𝐴 ∧ ω ⊆ 𝐴) ↔ (ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴))
13	1, 11, 12	3bitri 297	. . . . . 6 ⊢ (𝐴 ∈ (ran card ∖ ω) ↔ (ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴))
14	13	biimpi 216	. . . . 5 ⊢ (𝐴 ∈ (ran card ∖ ω) → (ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴))
15		cardalephex 10007	. . . . . . . 8 ⊢ (ω ⊆ 𝐴 → ((card‘𝐴) = 𝐴 ↔ ∃𝑥 ∈ On 𝐴 = (ℵ‘𝑥)))
16	15	biimpa 476	. . . . . . 7 ⊢ ((ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴) → ∃𝑥 ∈ On 𝐴 = (ℵ‘𝑥))
17		eqimss 3981	. . . . . . . . 9 ⊢ (𝐴 = (ℵ‘𝑥) → 𝐴 ⊆ (ℵ‘𝑥))
18	17	a1i 11	. . . . . . . 8 ⊢ ((ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴) → (𝐴 = (ℵ‘𝑥) → 𝐴 ⊆ (ℵ‘𝑥)))
19	18	reximdv 3153	. . . . . . 7 ⊢ ((ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴) → (∃𝑥 ∈ On 𝐴 = (ℵ‘𝑥) → ∃𝑥 ∈ On 𝐴 ⊆ (ℵ‘𝑥)))
20	16, 19	mpd 15	. . . . . 6 ⊢ ((ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴) → ∃𝑥 ∈ On 𝐴 ⊆ (ℵ‘𝑥))
21		onintrab2 7746	. . . . . 6 ⊢ (∃𝑥 ∈ On 𝐴 ⊆ (ℵ‘𝑥) ↔ ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∈ On)
22	20, 21	sylib 218	. . . . 5 ⊢ ((ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴) → ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∈ On)
23		simpr 484	. . . . . . 7 ⊢ (((ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴) ∧ ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∈ On) → ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∈ On)
24		onsuc 7759	. . . . . . 7 ⊢ (∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∈ On → suc ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∈ On)
25	23, 24	syl 17	. . . . . 6 ⊢ (((ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴) ∧ ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∈ On) → suc ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∈ On)
26		eloni 6329	. . . . . . . . 9 ⊢ (∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∈ On → Ord ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)})
27	23, 26	syl 17	. . . . . . . 8 ⊢ (((ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴) ∧ ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∈ On) → Ord ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)})
28		0elsuc 7781	. . . . . . . 8 ⊢ (Ord ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} → ∅ ∈ suc ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)})
29	27, 28	syl 17	. . . . . . 7 ⊢ (((ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴) ∧ ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∈ On) → ∅ ∈ suc ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)})
30		cardaleph 10006	. . . . . . . . 9 ⊢ ((ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴) → 𝐴 = (ℵ‘∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}))
31	30	adantr 480	. . . . . . . 8 ⊢ (((ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴) ∧ ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∈ On) → 𝐴 = (ℵ‘∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}))
32		sssucid 6401	. . . . . . . . 9 ⊢ ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ⊆ suc ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}
33		alephord3 9995	. . . . . . . . . 10 ⊢ ((∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∈ On ∧ suc ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∈ On) → (∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ⊆ suc ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ↔ (ℵ‘∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) ⊆ (ℵ‘suc ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)})))
34	23, 24, 33	syl2anc2 586	. . . . . . . . 9 ⊢ (((ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴) ∧ ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∈ On) → (∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ⊆ suc ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ↔ (ℵ‘∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) ⊆ (ℵ‘suc ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)})))
35	32, 34	mpbii 233	. . . . . . . 8 ⊢ (((ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴) ∧ ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∈ On) → (ℵ‘∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) ⊆ (ℵ‘suc ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}))
36	31, 35	eqsstrd 3957	. . . . . . 7 ⊢ (((ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴) ∧ ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∈ On) → 𝐴 ⊆ (ℵ‘suc ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}))
37		alephreg 10500	. . . . . . . 8 ⊢ (cf‘(ℵ‘suc ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)})) = (ℵ‘suc ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)})
38	37	a1i 11	. . . . . . 7 ⊢ (((ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴) ∧ ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∈ On) → (cf‘(ℵ‘suc ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)})) = (ℵ‘suc ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}))
39	29, 36, 38	3jca 1129	. . . . . 6 ⊢ (((ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴) ∧ ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∈ On) → (∅ ∈ suc ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∧ 𝐴 ⊆ (ℵ‘suc ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) ∧ (cf‘(ℵ‘suc ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)})) = (ℵ‘suc ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)})))
40	25, 39	jca 511	. . . . 5 ⊢ (((ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴) ∧ ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∈ On) → (suc ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∈ On ∧ (∅ ∈ suc ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∧ 𝐴 ⊆ (ℵ‘suc ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) ∧ (cf‘(ℵ‘suc ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)})) = (ℵ‘suc ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}))))
41	14, 22, 40	syl2anc2 586	. . . 4 ⊢ (𝐴 ∈ (ran card ∖ ω) → (suc ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∈ On ∧ (∅ ∈ suc ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∧ 𝐴 ⊆ (ℵ‘suc ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) ∧ (cf‘(ℵ‘suc ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)})) = (ℵ‘suc ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}))))
42	14, 16	syl 17	. . . . . . . 8 ⊢ (𝐴 ∈ (ran card ∖ ω) → ∃𝑥 ∈ On 𝐴 = (ℵ‘𝑥))
43	17	a1i 11	. . . . . . . . 9 ⊢ (𝐴 ∈ (ran card ∖ ω) → (𝐴 = (ℵ‘𝑥) → 𝐴 ⊆ (ℵ‘𝑥)))
44	43	reximdv 3153	. . . . . . . 8 ⊢ (𝐴 ∈ (ran card ∖ ω) → (∃𝑥 ∈ On 𝐴 = (ℵ‘𝑥) → ∃𝑥 ∈ On 𝐴 ⊆ (ℵ‘𝑥)))
45	42, 44	mpd 15	. . . . . . 7 ⊢ (𝐴 ∈ (ran card ∖ ω) → ∃𝑥 ∈ On 𝐴 ⊆ (ℵ‘𝑥))
46	45, 21	sylib 218	. . . . . 6 ⊢ (𝐴 ∈ (ran card ∖ ω) → ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∈ On)
47	46, 24	syl 17	. . . . 5 ⊢ (𝐴 ∈ (ran card ∖ ω) → suc ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∈ On)
48		sbcan 3779	. . . . . 6 ⊢ ([suc ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} / 𝑦](𝑦 ∈ On ∧ (∅ ∈ 𝑦 ∧ 𝐴 ⊆ (ℵ‘𝑦) ∧ (cf‘(ℵ‘𝑦)) = (ℵ‘𝑦))) ↔ ([suc ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} / 𝑦]𝑦 ∈ On ∧ [suc ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} / 𝑦](∅ ∈ 𝑦 ∧ 𝐴 ⊆ (ℵ‘𝑦) ∧ (cf‘(ℵ‘𝑦)) = (ℵ‘𝑦))))
49		sbcel1v 3795	. . . . . . . 8 ⊢ ([suc ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} / 𝑦]𝑦 ∈ On ↔ suc ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∈ On)
50	49	a1i 11	. . . . . . 7 ⊢ (suc ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∈ On → ([suc ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} / 𝑦]𝑦 ∈ On ↔ suc ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∈ On))
51		sbc3an 3794	. . . . . . . 8 ⊢ ([suc ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} / 𝑦](∅ ∈ 𝑦 ∧ 𝐴 ⊆ (ℵ‘𝑦) ∧ (cf‘(ℵ‘𝑦)) = (ℵ‘𝑦)) ↔ ([suc ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} / 𝑦]∅ ∈ 𝑦 ∧ [suc ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} / 𝑦]𝐴 ⊆ (ℵ‘𝑦) ∧ [suc ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} / 𝑦](cf‘(ℵ‘𝑦)) = (ℵ‘𝑦)))
52		sbcel2gv 3796	. . . . . . . . 9 ⊢ (suc ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∈ On → ([suc ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} / 𝑦]∅ ∈ 𝑦 ↔ ∅ ∈ suc ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}))
53		sbcssg 4462	. . . . . . . . . 10 ⊢ (suc ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∈ On → ([suc ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} / 𝑦]𝐴 ⊆ (ℵ‘𝑦) ↔ ⦋suc ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} / 𝑦⦌𝐴 ⊆ ⦋suc ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} / 𝑦⦌(ℵ‘𝑦)))
54		csbconstg 3857	. . . . . . . . . . 11 ⊢ (suc ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∈ On → ⦋suc ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} / 𝑦⦌𝐴 = 𝐴)
55		csbfv2g 6882	. . . . . . . . . . . 12 ⊢ (suc ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∈ On → ⦋suc ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} / 𝑦⦌(ℵ‘𝑦) = (ℵ‘⦋suc ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} / 𝑦⦌𝑦))
56		csbvarg 4375	. . . . . . . . . . . . 13 ⊢ (suc ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∈ On → ⦋suc ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} / 𝑦⦌𝑦 = suc ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)})
57	56	fveq2d 6840	. . . . . . . . . . . 12 ⊢ (suc ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∈ On → (ℵ‘⦋suc ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} / 𝑦⦌𝑦) = (ℵ‘suc ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}))
58	55, 57	eqtrd 2772	. . . . . . . . . . 11 ⊢ (suc ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∈ On → ⦋suc ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} / 𝑦⦌(ℵ‘𝑦) = (ℵ‘suc ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}))
59	54, 58	sseq12d 3956	. . . . . . . . . 10 ⊢ (suc ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∈ On → (⦋suc ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} / 𝑦⦌𝐴 ⊆ ⦋suc ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} / 𝑦⦌(ℵ‘𝑦) ↔ 𝐴 ⊆ (ℵ‘suc ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)})))
60	53, 59	bitrd 279	. . . . . . . . 9 ⊢ (suc ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∈ On → ([suc ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} / 𝑦]𝐴 ⊆ (ℵ‘𝑦) ↔ 𝐴 ⊆ (ℵ‘suc ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)})))
61		sbceqg 4353	. . . . . . . . . 10 ⊢ (suc ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∈ On → ([suc ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} / 𝑦](cf‘(ℵ‘𝑦)) = (ℵ‘𝑦) ↔ ⦋suc ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} / 𝑦⦌(cf‘(ℵ‘𝑦)) = ⦋suc ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} / 𝑦⦌(ℵ‘𝑦)))
62		csbfv2g 6882	. . . . . . . . . . . 12 ⊢ (suc ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∈ On → ⦋suc ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} / 𝑦⦌(cf‘(ℵ‘𝑦)) = (cf‘⦋suc ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} / 𝑦⦌(ℵ‘𝑦)))
63	58	fveq2d 6840	. . . . . . . . . . . 12 ⊢ (suc ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∈ On → (cf‘⦋suc ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} / 𝑦⦌(ℵ‘𝑦)) = (cf‘(ℵ‘suc ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)})))
64	62, 63	eqtrd 2772	. . . . . . . . . . 11 ⊢ (suc ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∈ On → ⦋suc ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} / 𝑦⦌(cf‘(ℵ‘𝑦)) = (cf‘(ℵ‘suc ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)})))
65	64, 58	eqeq12d 2753	. . . . . . . . . 10 ⊢ (suc ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∈ On → (⦋suc ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} / 𝑦⦌(cf‘(ℵ‘𝑦)) = ⦋suc ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} / 𝑦⦌(ℵ‘𝑦) ↔ (cf‘(ℵ‘suc ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)})) = (ℵ‘suc ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)})))
66	61, 65	bitrd 279	. . . . . . . . 9 ⊢ (suc ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∈ On → ([suc ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} / 𝑦](cf‘(ℵ‘𝑦)) = (ℵ‘𝑦) ↔ (cf‘(ℵ‘suc ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)})) = (ℵ‘suc ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)})))
67	52, 60, 66	3anbi123d 1439	. . . . . . . 8 ⊢ (suc ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∈ On → (([suc ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} / 𝑦]∅ ∈ 𝑦 ∧ [suc ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} / 𝑦]𝐴 ⊆ (ℵ‘𝑦) ∧ [suc ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} / 𝑦](cf‘(ℵ‘𝑦)) = (ℵ‘𝑦)) ↔ (∅ ∈ suc ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∧ 𝐴 ⊆ (ℵ‘suc ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) ∧ (cf‘(ℵ‘suc ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)})) = (ℵ‘suc ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}))))
68	51, 67	bitrid 283	. . . . . . 7 ⊢ (suc ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∈ On → ([suc ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} / 𝑦](∅ ∈ 𝑦 ∧ 𝐴 ⊆ (ℵ‘𝑦) ∧ (cf‘(ℵ‘𝑦)) = (ℵ‘𝑦)) ↔ (∅ ∈ suc ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∧ 𝐴 ⊆ (ℵ‘suc ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) ∧ (cf‘(ℵ‘suc ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)})) = (ℵ‘suc ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}))))
69	50, 68	anbi12d 633	. . . . . 6 ⊢ (suc ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∈ On → (([suc ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} / 𝑦]𝑦 ∈ On ∧ [suc ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} / 𝑦](∅ ∈ 𝑦 ∧ 𝐴 ⊆ (ℵ‘𝑦) ∧ (cf‘(ℵ‘𝑦)) = (ℵ‘𝑦))) ↔ (suc ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∈ On ∧ (∅ ∈ suc ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∧ 𝐴 ⊆ (ℵ‘suc ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) ∧ (cf‘(ℵ‘suc ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)})) = (ℵ‘suc ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)})))))
70	48, 69	bitrid 283	. . . . 5 ⊢ (suc ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∈ On → ([suc ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} / 𝑦](𝑦 ∈ On ∧ (∅ ∈ 𝑦 ∧ 𝐴 ⊆ (ℵ‘𝑦) ∧ (cf‘(ℵ‘𝑦)) = (ℵ‘𝑦))) ↔ (suc ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∈ On ∧ (∅ ∈ suc ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∧ 𝐴 ⊆ (ℵ‘suc ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) ∧ (cf‘(ℵ‘suc ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)})) = (ℵ‘suc ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)})))))
71	47, 70	syl 17	. . . 4 ⊢ (𝐴 ∈ (ran card ∖ ω) → ([suc ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} / 𝑦](𝑦 ∈ On ∧ (∅ ∈ 𝑦 ∧ 𝐴 ⊆ (ℵ‘𝑦) ∧ (cf‘(ℵ‘𝑦)) = (ℵ‘𝑦))) ↔ (suc ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∈ On ∧ (∅ ∈ suc ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∧ 𝐴 ⊆ (ℵ‘suc ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) ∧ (cf‘(ℵ‘suc ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)})) = (ℵ‘suc ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)})))))
72	41, 71	mpbird 257	. . 3 ⊢ (𝐴 ∈ (ran card ∖ ω) → [suc ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} / 𝑦](𝑦 ∈ On ∧ (∅ ∈ 𝑦 ∧ 𝐴 ⊆ (ℵ‘𝑦) ∧ (cf‘(ℵ‘𝑦)) = (ℵ‘𝑦))))
73	72	spesbcd 3822	. 2 ⊢ (𝐴 ∈ (ran card ∖ ω) → ∃𝑦(𝑦 ∈ On ∧ (∅ ∈ 𝑦 ∧ 𝐴 ⊆ (ℵ‘𝑦) ∧ (cf‘(ℵ‘𝑦)) = (ℵ‘𝑦))))
74		onintrab2 7746	. . 3 ⊢ (∃𝑦 ∈ On (∅ ∈ 𝑦 ∧ 𝐴 ⊆ (ℵ‘𝑦) ∧ (cf‘(ℵ‘𝑦)) = (ℵ‘𝑦)) ↔ ∩ {𝑦 ∈ On ∣ (∅ ∈ 𝑦 ∧ 𝐴 ⊆ (ℵ‘𝑦) ∧ (cf‘(ℵ‘𝑦)) = (ℵ‘𝑦))} ∈ On)
75		df-rex 3063	. . 3 ⊢ (∃𝑦 ∈ On (∅ ∈ 𝑦 ∧ 𝐴 ⊆ (ℵ‘𝑦) ∧ (cf‘(ℵ‘𝑦)) = (ℵ‘𝑦)) ↔ ∃𝑦(𝑦 ∈ On ∧ (∅ ∈ 𝑦 ∧ 𝐴 ⊆ (ℵ‘𝑦) ∧ (cf‘(ℵ‘𝑦)) = (ℵ‘𝑦))))
76		risset 3213	. . 3 ⊢ (∩ {𝑦 ∈ On ∣ (∅ ∈ 𝑦 ∧ 𝐴 ⊆ (ℵ‘𝑦) ∧ (cf‘(ℵ‘𝑦)) = (ℵ‘𝑦))} ∈ On ↔ ∃𝑥 ∈ On 𝑥 = ∩ {𝑦 ∈ On ∣ (∅ ∈ 𝑦 ∧ 𝐴 ⊆ (ℵ‘𝑦) ∧ (cf‘(ℵ‘𝑦)) = (ℵ‘𝑦))})
77	74, 75, 76	3bitr3i 301	. 2 ⊢ (∃𝑦(𝑦 ∈ On ∧ (∅ ∈ 𝑦 ∧ 𝐴 ⊆ (ℵ‘𝑦) ∧ (cf‘(ℵ‘𝑦)) = (ℵ‘𝑦))) ↔ ∃𝑥 ∈ On 𝑥 = ∩ {𝑦 ∈ On ∣ (∅ ∈ 𝑦 ∧ 𝐴 ⊆ (ℵ‘𝑦) ∧ (cf‘(ℵ‘𝑦)) = (ℵ‘𝑦))})
78	73, 77	sylib 218	1 ⊢ (𝐴 ∈ (ran card ∖ ω) → ∃𝑥 ∈ On 𝑥 = ∩ {𝑦 ∈ On ∣ (∅ ∈ 𝑦 ∧ 𝐴 ⊆ (ℵ‘𝑦) ∧ (cf‘(ℵ‘𝑦)) = (ℵ‘𝑦))})

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542  ∃wex 1781   ∈ wcel 2114  ∃wrex 3062  {crab 3390  [wsbc 3729  ⦋csb 3838   ∖ cdif 3887   ⊆ wss 3890  ∅c0 4274  ∩ cint 4890  ran crn 5627  Ord word 6318  Oncon0 6319  suc csuc 6321  ‘cfv 6494  ωcom 7812  cardccrd 9854  ℵcale 9855  cfccf 9856

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pow 5304  ax-pr 5372  ax-un 7684  ax-inf2 9557  ax-ac2 10380

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5521  df-eprel 5526  df-po 5534  df-so 5535  df-fr 5579  df-se 5580  df-we 5581  df-xp 5632  df-rel 5633  df-cnv 5634  df-co 5635  df-dm 5636  df-rn 5637  df-res 5638  df-ima 5639  df-pred 6261  df-ord 6322  df-on 6323  df-lim 6324  df-suc 6325  df-iota 6450  df-fun 6496  df-fn 6497  df-f 6498  df-f1 6499  df-fo 6500  df-f1o 6501  df-fv 6502  df-isom 6503  df-riota 7319  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7813  df-1st 7937  df-2nd 7938  df-frecs 8226  df-wrecs 8257  df-recs 8306  df-rdg 8344  df-1o 8400  df-er 8638  df-map 8770  df-en 8889  df-dom 8890  df-sdom 8891  df-fin 8892  df-oi 9420  df-har 9467  df-card 9858  df-aleph 9859  df-cf 9860  df-acn 9861  df-ac 10033

This theorem is referenced by:  minregex2  43984