Theorem minregex 44186

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 3923	. . . . . . 7 ⊢ (𝐴 ∈ (ran card ∖ ω) ↔ (𝐴 ∈ ran card ∧ ¬ 𝐴 ∈ ω))
2		omelon 9615	. . . . . . . . . 10 ⊢ ω ∈ On
3		cardon 9930	. . . . . . . . . . 11 ⊢ (card‘𝐴) ∈ On
4		eleq1 2857	. . . . . . . . . . 11 ⊢ ((card‘𝐴) = 𝐴 → ((card‘𝐴) ∈ On ↔ 𝐴 ∈ On))
5	3, 4	mpbii 236	. . . . . . . . . 10 ⊢ ((card‘𝐴) = 𝐴 → 𝐴 ∈ On)
6		ontri1 6396	. . . . . . . . . 10 ⊢ ((ω ∈ On ∧ 𝐴 ∈ On) → (ω ⊆ 𝐴 ↔ ¬ 𝐴 ∈ ω))
7	2, 5, 6	sylancr 598	. . . . . . . . 9 ⊢ ((card‘𝐴) = 𝐴 → (ω ⊆ 𝐴 ↔ ¬ 𝐴 ∈ ω))
8	7	pm5.32i 584	. . . . . . . 8 ⊢ (((card‘𝐴) = 𝐴 ∧ ω ⊆ 𝐴) ↔ ((card‘𝐴) = 𝐴 ∧ ¬ 𝐴 ∈ ω))
9		iscard4 44185	. . . . . . . . 9 ⊢ ((card‘𝐴) = 𝐴 ↔ 𝐴 ∈ ran card)
10	9	anbi1i 635	. . . . . . . 8 ⊢ (((card‘𝐴) = 𝐴 ∧ ¬ 𝐴 ∈ ω) ↔ (𝐴 ∈ ran card ∧ ¬ 𝐴 ∈ ω))
11	8, 10	bitr2i 279	. . . . . . 7 ⊢ ((𝐴 ∈ ran card ∧ ¬ 𝐴 ∈ ω) ↔ ((card‘𝐴) = 𝐴 ∧ ω ⊆ 𝐴))
12		ancom 465	. . . . . . 7 ⊢ (((card‘𝐴) = 𝐴 ∧ ω ⊆ 𝐴) ↔ (ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴))
13	1, 11, 12	3bitri 300	. . . . . 6 ⊢ (𝐴 ∈ (ran card ∖ ω) ↔ (ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴))
14	13	biimpi 219	. . . . 5 ⊢ (𝐴 ∈ (ran card ∖ ω) → (ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴))
15		cardalephex 10074	. . . . . . . 8 ⊢ (ω ⊆ 𝐴 → ((card‘𝐴) = 𝐴 ↔ ∃𝑥 ∈ On 𝐴 = (ℵ‘𝑥)))
16	15	biimpa 481	. . . . . . 7 ⊢ ((ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴) → ∃𝑥 ∈ On 𝐴 = (ℵ‘𝑥))
17		eqimss 4003	. . . . . . . . 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 3186	. . . . . . 7 ⊢ ((ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴) → (∃𝑥 ∈ On 𝐴 = (ℵ‘𝑥) → ∃𝑥 ∈ On 𝐴 ⊆ (ℵ‘𝑥)))
20	16, 19	mpd 16	. . . . . 6 ⊢ ((ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴) → ∃𝑥 ∈ On 𝐴 ⊆ (ℵ‘𝑥))
21		onintrab2 7796	. . . . . 6 ⊢ (∃𝑥 ∈ On 𝐴 ⊆ (ℵ‘𝑥) ↔ ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∈ On)
22	20, 21	sylib 221	. . . . 5 ⊢ ((ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴) → ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∈ On)
23		simpr 489	. . . . . . 7 ⊢ (((ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴) ∧ ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∈ On) → ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∈ On)
24		onsuc 7809	. . . . . . 7 ⊢ (∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∈ On → suc ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∈ On)
25	23, 24	syl 18	. . . . . 6 ⊢ (((ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴) ∧ ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∈ On) → suc ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∈ On)
26		eloni 6371	. . . . . . . . 9 ⊢ (∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∈ On → Ord ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)})
27	23, 26	syl 18	. . . . . . . 8 ⊢ (((ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴) ∧ ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∈ On) → Ord ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)})
28		0elsuc 7831	. . . . . . . 8 ⊢ (Ord ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} → ∅ ∈ suc ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)})
29	27, 28	syl 18	. . . . . . 7 ⊢ (((ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴) ∧ ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∈ On) → ∅ ∈ suc ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)})
30		cardaleph 10073	. . . . . . . . 9 ⊢ ((ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴) → 𝐴 = (ℵ‘∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}))
31	30	adantr 485	. . . . . . . 8 ⊢ (((ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴) ∧ ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∈ On) → 𝐴 = (ℵ‘∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}))
32		sssucid 6444	. . . . . . . . 9 ⊢ ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ⊆ suc ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}
33		alephord3 10062	. . . . . . . . . 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 596	. . . . . . . . 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 236	. . . . . . . 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 3979	. . . . . . 7 ⊢ (((ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴) ∧ ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∈ On) → 𝐴 ⊆ (ℵ‘suc ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}))
37		alephreg 10567	. . . . . . . 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 1144	. . . . . 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 520	. . . . 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 596	. . . 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 18	. . . . . . . 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 3186	. . . . . . . 8 ⊢ (𝐴 ∈ (ran card ∖ ω) → (∃𝑥 ∈ On 𝐴 = (ℵ‘𝑥) → ∃𝑥 ∈ On 𝐴 ⊆ (ℵ‘𝑥)))
45	42, 44	mpd 16	. . . . . . 7 ⊢ (𝐴 ∈ (ran card ∖ ω) → ∃𝑥 ∈ On 𝐴 ⊆ (ℵ‘𝑥))
46	45, 21	sylib 221	. . . . . 6 ⊢ (𝐴 ∈ (ran card ∖ ω) → ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∈ On)
47	46, 24	syl 18	. . . . 5 ⊢ (𝐴 ∈ (ran card ∖ ω) → suc ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∈ On)
48		sbcan 3802	. . . . . 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 3818	. . . . . . . 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 3817	. . . . . . . 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 3819	. . . . . . . . 9 ⊢ (suc ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∈ On → ([suc ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} / 𝑦]∅ ∈ 𝑦 ↔ ∅ ∈ suc ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}))
53		sbcssg 4487	. . . . . . . . . 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 3880	. . . . . . . . . . 11 ⊢ (suc ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∈ On → ⦋suc ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} / 𝑦⦌𝐴 = 𝐴)
55		csbfv2g 6928	. . . . . . . . . . . 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 4405	. . . . . . . . . . . . 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 6886	. . . . . . . . . . . 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 2804	. . . . . . . . . . 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 3978	. . . . . . . . . 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 282	. . . . . . . . 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 4383	. . . . . . . . . 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 6928	. . . . . . . . . . . 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 6886	. . . . . . . . . . . 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 2804	. . . . . . . . . . 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 2785	. . . . . . . . . 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 282	. . . . . . . . 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 1462	. . . . . . . 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 286	. . . . . . 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 643	. . . . . 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 286	. . . . 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 18	. . . 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 260	. . 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 3845	. 2 ⊢ (𝐴 ∈ (ran card ∖ ω) → ∃𝑦(𝑦 ∈ On ∧ (∅ ∈ 𝑦 ∧ 𝐴 ⊆ (ℵ‘𝑦) ∧ (cf‘(ℵ‘𝑦)) = (ℵ‘𝑦))))
74		onintrab2 7796	. . 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 3096	. . 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 3246	. . 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 304	. 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 221	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 209   ∧ wa 400   ∧ w3a 1101   = wceq 1567  ∃wex 1806   ∈ wcel 2149  ∃wrex 3095  {crab 3423  [wsbc 3753  ⦋csb 3861   ∖ cdif 3910   ⊆ wss 3913  ∅c0 4294  ∩ cint 4916  ran crn 5663  Ord word 6360  Oncon0 6361  suc csuc 6363  ‘cfv 6537  ωcom 7862  cardccrd 9921  ℵcale 9922  cfccf 9923

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733  ax-inf2 9610  ax-ac2 10447

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-int 4917  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-se 5616  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-isom 6546  df-riota 7368  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7863  df-1st 7986  df-2nd 7987  df-frecs 8278  df-wrecs 8309  df-recs 8358  df-rdg 8397  df-1o 8453  df-er 8694  df-map 8826  df-en 8944  df-dom 8945  df-sdom 8946  df-fin 8947  df-oi 9472  df-har 9519  df-card 9925  df-aleph 9926  df-cf 9927  df-acn 9928  df-ac 10100

This theorem is referenced by:  minregex2  44187