Theorem minregex 43496

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 3986	. . . . . . 7 ⊢ (𝐴 ∈ (ran card ∖ ω) ↔ (𝐴 ∈ ran card ∧ ¬ 𝐴 ∈ ω))
2		omelon 9715	. . . . . . . . . 10 ⊢ ω ∈ On
3		cardon 10013	. . . . . . . . . . 11 ⊢ (card‘𝐴) ∈ On
4		eleq1 2832	. . . . . . . . . . 11 ⊢ ((card‘𝐴) = 𝐴 → ((card‘𝐴) ∈ On ↔ 𝐴 ∈ On))
5	3, 4	mpbii 233	. . . . . . . . . 10 ⊢ ((card‘𝐴) = 𝐴 → 𝐴 ∈ On)
6		ontri1 6429	. . . . . . . . . 10 ⊢ ((ω ∈ On ∧ 𝐴 ∈ On) → (ω ⊆ 𝐴 ↔ ¬ 𝐴 ∈ ω))
7	2, 5, 6	sylancr 586	. . . . . . . . 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 43495	. . . . . . . . 9 ⊢ ((card‘𝐴) = 𝐴 ↔ 𝐴 ∈ ran card)
10	9	anbi1i 623	. . . . . . . 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 10159	. . . . . . . 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 4067	. . . . . . . . 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 3176	. . . . . . 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 7833	. . . . . 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 7847	. . . . . . 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 6405	. . . . . . . . 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 7871	. . . . . . . 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 10158	. . . . . . . . 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 6475	. . . . . . . . 9 ⊢ ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ⊆ suc ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}
33		alephord3 10147	. . . . . . . . . 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 584	. . . . . . . . 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 4047	. . . . . . 7 ⊢ (((ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴) ∧ ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∈ On) → 𝐴 ⊆ (ℵ‘suc ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}))
37		alephreg 10651	. . . . . . . 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 584	. . . 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 3176	. . . . . . . 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 3857	. . . . . 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 3875	. . . . . . . 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 3874	. . . . . . . 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 3876	. . . . . . . . 9 ⊢ (suc ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∈ On → ([suc ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} / 𝑦]∅ ∈ 𝑦 ↔ ∅ ∈ suc ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}))
53		sbcssg 4543	. . . . . . . . . 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 3940	. . . . . . . . . . 11 ⊢ (suc ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∈ On → ⦋suc ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} / 𝑦⦌𝐴 = 𝐴)
55		csbfv2g 6969	. . . . . . . . . . . 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 4457	. . . . . . . . . . . . 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 6924	. . . . . . . . . . . 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 2780	. . . . . . . . . . 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 4042	. . . . . . . . . 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 4435	. . . . . . . . . 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 6969	. . . . . . . . . . . 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 6924	. . . . . . . . . . . 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 2780	. . . . . . . . . . 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 2756	. . . . . . . . . 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 1436	. . . . . . . 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 631	. . . . . 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 3905	. 2 ⊢ (𝐴 ∈ (ran card ∖ ω) → ∃𝑦(𝑦 ∈ On ∧ (∅ ∈ 𝑦 ∧ 𝐴 ⊆ (ℵ‘𝑦) ∧ (cf‘(ℵ‘𝑦)) = (ℵ‘𝑦))))
74		onintrab2 7833	. . 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 3077	. . 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 3239	. . 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 1537  ∃wex 1777   ∈ wcel 2108  ∃wrex 3076  {crab 3443  [wsbc 3804  ⦋csb 3921   ∖ cdif 3973   ⊆ wss 3976  ∅c0 4352  ∩ cint 4970  ran crn 5701  Ord word 6394  Oncon0 6395  suc csuc 6397  ‘cfv 6573  ωcom 7903  cardccrd 10004  ℵcale 10005  cfccf 10006

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-inf2 9710  ax-ac2 10532

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-se 5653  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-isom 6582  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-er 8763  df-map 8886  df-en 9004  df-dom 9005  df-sdom 9006  df-fin 9007  df-oi 9579  df-har 9626  df-card 10008  df-aleph 10009  df-cf 10010  df-acn 10011  df-ac 10185

This theorem is referenced by:  minregex2  43497