Theorem minregex 43516

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 3926	. . . . . . 7 ⊢ (𝐴 ∈ (ran card ∖ ω) ↔ (𝐴 ∈ ran card ∧ ¬ 𝐴 ∈ ω))
2		omelon 9605	. . . . . . . . . 10 ⊢ ω ∈ On
3		cardon 9903	. . . . . . . . . . 11 ⊢ (card‘𝐴) ∈ On
4		eleq1 2817	. . . . . . . . . . 11 ⊢ ((card‘𝐴) = 𝐴 → ((card‘𝐴) ∈ On ↔ 𝐴 ∈ On))
5	3, 4	mpbii 233	. . . . . . . . . 10 ⊢ ((card‘𝐴) = 𝐴 → 𝐴 ∈ On)
6		ontri1 6368	. . . . . . . . . 10 ⊢ ((ω ∈ On ∧ 𝐴 ∈ On) → (ω ⊆ 𝐴 ↔ ¬ 𝐴 ∈ ω))
7	2, 5, 6	sylancr 587	. . . . . . . . 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 43515	. . . . . . . . 9 ⊢ ((card‘𝐴) = 𝐴 ↔ 𝐴 ∈ ran card)
10	9	anbi1i 624	. . . . . . . 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 10049	. . . . . . . 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 4007	. . . . . . . . 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 3149	. . . . . . 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 7775	. . . . . 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 7789	. . . . . . 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 6344	. . . . . . . . 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 7812	. . . . . . . 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 10048	. . . . . . . . 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 6416	. . . . . . . . 9 ⊢ ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ⊆ suc ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}
33		alephord3 10037	. . . . . . . . . 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 585	. . . . . . . . 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 3983	. . . . . . 7 ⊢ (((ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴) ∧ ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∈ On) → 𝐴 ⊆ (ℵ‘suc ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}))
37		alephreg 10541	. . . . . . . 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 1128	. . . . . 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 585	. . . 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 3149	. . . . . . . 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 3805	. . . . . 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 3821	. . . . . . . 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 3820	. . . . . . . 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 3822	. . . . . . . . 9 ⊢ (suc ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∈ On → ([suc ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} / 𝑦]∅ ∈ 𝑦 ↔ ∅ ∈ suc ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}))
53		sbcssg 4485	. . . . . . . . . 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 3883	. . . . . . . . . . 11 ⊢ (suc ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∈ On → ⦋suc ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} / 𝑦⦌𝐴 = 𝐴)
55		csbfv2g 6909	. . . . . . . . . . . 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 4399	. . . . . . . . . . . . 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 6864	. . . . . . . . . . . 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 2765	. . . . . . . . . . 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 3982	. . . . . . . . . 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 4377	. . . . . . . . . 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 6909	. . . . . . . . . . . 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 6864	. . . . . . . . . . . 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 2765	. . . . . . . . . . 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 2746	. . . . . . . . . 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 1438	. . . . . . . 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 632	. . . . . 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 3848	. 2 ⊢ (𝐴 ∈ (ran card ∖ ω) → ∃𝑦(𝑦 ∈ On ∧ (∅ ∈ 𝑦 ∧ 𝐴 ⊆ (ℵ‘𝑦) ∧ (cf‘(ℵ‘𝑦)) = (ℵ‘𝑦))))
74		onintrab2 7775	. . 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 3055	. . 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 1086   = wceq 1540  ∃wex 1779   ∈ wcel 2109  ∃wrex 3054  {crab 3408  [wsbc 3755  ⦋csb 3864   ∖ cdif 3913   ⊆ wss 3916  ∅c0 4298  ∩ cint 4912  ran crn 5641  Ord word 6333  Oncon0 6334  suc csuc 6336  ‘cfv 6513  ωcom 7844  cardccrd 9894  ℵcale 9895  cfccf 9896

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5236  ax-sep 5253  ax-nul 5263  ax-pow 5322  ax-pr 5389  ax-un 7713  ax-inf2 9600  ax-ac2 10422

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 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3756  df-csb 3865  df-dif 3919  df-un 3921  df-in 3923  df-ss 3933  df-pss 3936  df-nul 4299  df-if 4491  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-int 4913  df-iun 4959  df-br 5110  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5535  df-eprel 5540  df-po 5548  df-so 5549  df-fr 5593  df-se 5594  df-we 5595  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-rn 5651  df-res 5652  df-ima 5653  df-pred 6276  df-ord 6337  df-on 6338  df-lim 6339  df-suc 6340  df-iota 6466  df-fun 6515  df-fn 6516  df-f 6517  df-f1 6518  df-fo 6519  df-f1o 6520  df-fv 6521  df-isom 6522  df-riota 7346  df-ov 7392  df-oprab 7393  df-mpo 7394  df-om 7845  df-1st 7970  df-2nd 7971  df-frecs 8262  df-wrecs 8293  df-recs 8342  df-rdg 8380  df-1o 8436  df-er 8673  df-map 8803  df-en 8921  df-dom 8922  df-sdom 8923  df-fin 8924  df-oi 9469  df-har 9516  df-card 9898  df-aleph 9899  df-cf 9900  df-acn 9901  df-ac 10075

This theorem is referenced by:  minregex2  43517