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Theorem pwcfsdom 10519

Description: A corollary of Konig's Theorem konigth 10505. Theorem 11.28 of [TakeutiZaring] p. 108. (Contributed by Mario Carneiro, 20-Mar-2013.)

**Hypothesis**
Ref	Expression
pwcfsdom.1	⊢ 𝐻 = (𝑦 ∈ (cf‘(ℵ‘𝐴)) ↦ (har‘(𝑓‘𝑦)))

**Assertion**
Ref	Expression
pwcfsdom	⊢ (ℵ‘𝐴) ≺ ((ℵ‘𝐴) ↑_m (cf‘(ℵ‘𝐴)))

Distinct variable group: 𝐴,𝑓,𝑦 Allowed substitution hints: 𝐻(𝑦,𝑓)

Proof of Theorem pwcfsdom Dummy variables 𝑤 𝑧 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		onzsl 7782	. . . 4 ⊢ (𝐴 ∈ On ↔ (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ (𝐴 ∈ V ∧ Lim 𝐴)))
2	1	biimpi 215	. . 3 ⊢ (𝐴 ∈ On → (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ (𝐴 ∈ V ∧ Lim 𝐴)))
3		cfom 10200	. . . . . . 7 ⊢ (cf‘ω) = ω
4		aleph0 10002	. . . . . . . 8 ⊢ (ℵ‘∅) = ω
5	4	fveq2i 6845	. . . . . . 7 ⊢ (cf‘(ℵ‘∅)) = (cf‘ω)
6	3, 5, 4	3eqtr4i 2774	. . . . . 6 ⊢ (cf‘(ℵ‘∅)) = (ℵ‘∅)
7		2fveq3 6847	. . . . . 6 ⊢ (𝐴 = ∅ → (cf‘(ℵ‘𝐴)) = (cf‘(ℵ‘∅)))
8		fveq2 6842	. . . . . 6 ⊢ (𝐴 = ∅ → (ℵ‘𝐴) = (ℵ‘∅))
9	6, 7, 8	3eqtr4a 2802	. . . . 5 ⊢ (𝐴 = ∅ → (cf‘(ℵ‘𝐴)) = (ℵ‘𝐴))
10		fvex 6855	. . . . . . . . 9 ⊢ (ℵ‘𝐴) ∈ V
11	10	canth2 9074	. . . . . . . 8 ⊢ (ℵ‘𝐴) ≺ 𝒫 (ℵ‘𝐴)
12	10	pw2en 9023	. . . . . . . 8 ⊢ 𝒫 (ℵ‘𝐴) ≈ (2_o ↑_m (ℵ‘𝐴))
13		sdomentr 9055	. . . . . . . 8 ⊢ (((ℵ‘𝐴) ≺ 𝒫 (ℵ‘𝐴) ∧ 𝒫 (ℵ‘𝐴) ≈ (2_o ↑_m (ℵ‘𝐴))) → (ℵ‘𝐴) ≺ (2_o ↑_m (ℵ‘𝐴)))
14	11, 12, 13	mp2an 690	. . . . . . 7 ⊢ (ℵ‘𝐴) ≺ (2_o ↑_m (ℵ‘𝐴))
15		alephon 10005	. . . . . . . . 9 ⊢ (ℵ‘𝐴) ∈ On
16		alephgeom 10018	. . . . . . . . . 10 ⊢ (𝐴 ∈ On ↔ ω ⊆ (ℵ‘𝐴))
17		omelon 9582	. . . . . . . . . . . 12 ⊢ ω ∈ On
18		2onn 8588	. . . . . . . . . . . 12 ⊢ 2_o ∈ ω
19		onelss 6359	. . . . . . . . . . . 12 ⊢ (ω ∈ On → (2_o ∈ ω → 2_o ⊆ ω))
20	17, 18, 19	mp2 9	. . . . . . . . . . 11 ⊢ 2_o ⊆ ω
21		sstr 3952	. . . . . . . . . . 11 ⊢ ((2_o ⊆ ω ∧ ω ⊆ (ℵ‘𝐴)) → 2_o ⊆ (ℵ‘𝐴))
22	20, 21	mpan 688	. . . . . . . . . 10 ⊢ (ω ⊆ (ℵ‘𝐴) → 2_o ⊆ (ℵ‘𝐴))
23	16, 22	sylbi 216	. . . . . . . . 9 ⊢ (𝐴 ∈ On → 2_o ⊆ (ℵ‘𝐴))
24		ssdomg 8940	. . . . . . . . 9 ⊢ ((ℵ‘𝐴) ∈ On → (2_o ⊆ (ℵ‘𝐴) → 2_o ≼ (ℵ‘𝐴)))
25	15, 23, 24	mpsyl 68	. . . . . . . 8 ⊢ (𝐴 ∈ On → 2_o ≼ (ℵ‘𝐴))
26		mapdom1 9086	. . . . . . . 8 ⊢ (2_o ≼ (ℵ‘𝐴) → (2_o ↑_m (ℵ‘𝐴)) ≼ ((ℵ‘𝐴) ↑_m (ℵ‘𝐴)))
27	25, 26	syl 17	. . . . . . 7 ⊢ (𝐴 ∈ On → (2_o ↑_m (ℵ‘𝐴)) ≼ ((ℵ‘𝐴) ↑_m (ℵ‘𝐴)))
28		sdomdomtr 9054	. . . . . . 7 ⊢ (((ℵ‘𝐴) ≺ (2_o ↑_m (ℵ‘𝐴)) ∧ (2_o ↑_m (ℵ‘𝐴)) ≼ ((ℵ‘𝐴) ↑_m (ℵ‘𝐴))) → (ℵ‘𝐴) ≺ ((ℵ‘𝐴) ↑_m (ℵ‘𝐴)))
29	14, 27, 28	sylancr 587	. . . . . 6 ⊢ (𝐴 ∈ On → (ℵ‘𝐴) ≺ ((ℵ‘𝐴) ↑_m (ℵ‘𝐴)))
30		oveq2 7365	. . . . . . 7 ⊢ ((cf‘(ℵ‘𝐴)) = (ℵ‘𝐴) → ((ℵ‘𝐴) ↑_m (cf‘(ℵ‘𝐴))) = ((ℵ‘𝐴) ↑_m (ℵ‘𝐴)))
31	30	breq2d 5117	. . . . . 6 ⊢ ((cf‘(ℵ‘𝐴)) = (ℵ‘𝐴) → ((ℵ‘𝐴) ≺ ((ℵ‘𝐴) ↑_m (cf‘(ℵ‘𝐴))) ↔ (ℵ‘𝐴) ≺ ((ℵ‘𝐴) ↑_m (ℵ‘𝐴))))
32	29, 31	syl5ibrcom 246	. . . . 5 ⊢ (𝐴 ∈ On → ((cf‘(ℵ‘𝐴)) = (ℵ‘𝐴) → (ℵ‘𝐴) ≺ ((ℵ‘𝐴) ↑_m (cf‘(ℵ‘𝐴)))))
33	9, 32	syl5 34	. . . 4 ⊢ (𝐴 ∈ On → (𝐴 = ∅ → (ℵ‘𝐴) ≺ ((ℵ‘𝐴) ↑_m (cf‘(ℵ‘𝐴)))))
34		alephreg 10518	. . . . . . 7 ⊢ (cf‘(ℵ‘suc 𝑥)) = (ℵ‘suc 𝑥)
35		2fveq3 6847	. . . . . . 7 ⊢ (𝐴 = suc 𝑥 → (cf‘(ℵ‘𝐴)) = (cf‘(ℵ‘suc 𝑥)))
36		fveq2 6842	. . . . . . 7 ⊢ (𝐴 = suc 𝑥 → (ℵ‘𝐴) = (ℵ‘suc 𝑥))
37	34, 35, 36	3eqtr4a 2802	. . . . . 6 ⊢ (𝐴 = suc 𝑥 → (cf‘(ℵ‘𝐴)) = (ℵ‘𝐴))
38	37	rexlimivw 3148	. . . . 5 ⊢ (∃𝑥 ∈ On 𝐴 = suc 𝑥 → (cf‘(ℵ‘𝐴)) = (ℵ‘𝐴))
39	38, 32	syl5 34	. . . 4 ⊢ (𝐴 ∈ On → (∃𝑥 ∈ On 𝐴 = suc 𝑥 → (ℵ‘𝐴) ≺ ((ℵ‘𝐴) ↑_m (cf‘(ℵ‘𝐴)))))
40		cfsmo 10207	. . . . . 6 ⊢ ((ℵ‘𝐴) ∈ On → ∃𝑓(𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) ∧ Smo 𝑓 ∧ ∀𝑧 ∈ (ℵ‘𝐴)∃𝑤 ∈ (cf‘(ℵ‘𝐴))𝑧 ⊆ (𝑓‘𝑤)))
41		limelon 6381	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ V ∧ Lim 𝐴) → 𝐴 ∈ On)
42		ffn 6668	. . . . . . . . . . . . . . . 16 ⊢ (𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) → 𝑓 Fn (cf‘(ℵ‘𝐴)))
43		fnrnfv 6902	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 Fn (cf‘(ℵ‘𝐴)) → ran 𝑓 = {𝑦 ∣ ∃𝑥 ∈ (cf‘(ℵ‘𝐴))𝑦 = (𝑓‘𝑥)})
44	43	unieqd 4879	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 Fn (cf‘(ℵ‘𝐴)) → ∪ ran 𝑓 = ∪ {𝑦 ∣ ∃𝑥 ∈ (cf‘(ℵ‘𝐴))𝑦 = (𝑓‘𝑥)})
45	42, 44	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) → ∪ ran 𝑓 = ∪ {𝑦 ∣ ∃𝑥 ∈ (cf‘(ℵ‘𝐴))𝑦 = (𝑓‘𝑥)})
46		fvex 6855	. . . . . . . . . . . . . . . 16 ⊢ (𝑓‘𝑥) ∈ V
47	46	dfiun2 4993	. . . . . . . . . . . . . . 15 ⊢ ∪ 𝑥 ∈ (cf‘(ℵ‘𝐴))(𝑓‘𝑥) = ∪ {𝑦 ∣ ∃𝑥 ∈ (cf‘(ℵ‘𝐴))𝑦 = (𝑓‘𝑥)}
48	45, 47	eqtr4di 2794	. . . . . . . . . . . . . 14 ⊢ (𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) → ∪ ran 𝑓 = ∪ 𝑥 ∈ (cf‘(ℵ‘𝐴))(𝑓‘𝑥))
49	48	ad2antrl 726	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ On ∧ (𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) ∧ ∀𝑧 ∈ (ℵ‘𝐴)∃𝑤 ∈ (cf‘(ℵ‘𝐴))𝑧 ⊆ (𝑓‘𝑤))) → ∪ ran 𝑓 = ∪ 𝑥 ∈ (cf‘(ℵ‘𝐴))(𝑓‘𝑥))
50		fnfvelrn 7031	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑓 Fn (cf‘(ℵ‘𝐴)) ∧ 𝑤 ∈ (cf‘(ℵ‘𝐴))) → (𝑓‘𝑤) ∈ ran 𝑓)
51	42, 50	sylan 580	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) ∧ 𝑤 ∈ (cf‘(ℵ‘𝐴))) → (𝑓‘𝑤) ∈ ran 𝑓)
52		sseq2 3970	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = (𝑓‘𝑤) → (𝑧 ⊆ 𝑦 ↔ 𝑧 ⊆ (𝑓‘𝑤)))
53	52	rspcev 3581	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑓‘𝑤) ∈ ran 𝑓 ∧ 𝑧 ⊆ (𝑓‘𝑤)) → ∃𝑦 ∈ ran 𝑓 𝑧 ⊆ 𝑦)
54	51, 53	sylan 580	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) ∧ 𝑤 ∈ (cf‘(ℵ‘𝐴))) ∧ 𝑧 ⊆ (𝑓‘𝑤)) → ∃𝑦 ∈ ran 𝑓 𝑧 ⊆ 𝑦)
55	54	rexlimdva2 3154	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) → (∃𝑤 ∈ (cf‘(ℵ‘𝐴))𝑧 ⊆ (𝑓‘𝑤) → ∃𝑦 ∈ ran 𝑓 𝑧 ⊆ 𝑦))
56	55	ralimdv 3166	. . . . . . . . . . . . . . . 16 ⊢ (𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) → (∀𝑧 ∈ (ℵ‘𝐴)∃𝑤 ∈ (cf‘(ℵ‘𝐴))𝑧 ⊆ (𝑓‘𝑤) → ∀𝑧 ∈ (ℵ‘𝐴)∃𝑦 ∈ ran 𝑓 𝑧 ⊆ 𝑦))
57	56	imp 407	. . . . . . . . . . . . . . 15 ⊢ ((𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) ∧ ∀𝑧 ∈ (ℵ‘𝐴)∃𝑤 ∈ (cf‘(ℵ‘𝐴))𝑧 ⊆ (𝑓‘𝑤)) → ∀𝑧 ∈ (ℵ‘𝐴)∃𝑦 ∈ ran 𝑓 𝑧 ⊆ 𝑦)
58	57	adantl 482	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ On ∧ (𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) ∧ ∀𝑧 ∈ (ℵ‘𝐴)∃𝑤 ∈ (cf‘(ℵ‘𝐴))𝑧 ⊆ (𝑓‘𝑤))) → ∀𝑧 ∈ (ℵ‘𝐴)∃𝑦 ∈ ran 𝑓 𝑧 ⊆ 𝑦)
59		alephislim 10019	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ On ↔ Lim (ℵ‘𝐴))
60	59	biimpi 215	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ On → Lim (ℵ‘𝐴))
61		frn 6675	. . . . . . . . . . . . . . . 16 ⊢ (𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) → ran 𝑓 ⊆ (ℵ‘𝐴))
62	61	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) ∧ ∀𝑧 ∈ (ℵ‘𝐴)∃𝑤 ∈ (cf‘(ℵ‘𝐴))𝑧 ⊆ (𝑓‘𝑤)) → ran 𝑓 ⊆ (ℵ‘𝐴))
63		coflim 10197	. . . . . . . . . . . . . . 15 ⊢ ((Lim (ℵ‘𝐴) ∧ ran 𝑓 ⊆ (ℵ‘𝐴)) → (∪ ran 𝑓 = (ℵ‘𝐴) ↔ ∀𝑧 ∈ (ℵ‘𝐴)∃𝑦 ∈ ran 𝑓 𝑧 ⊆ 𝑦))
64	60, 62, 63	syl2an 596	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ On ∧ (𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) ∧ ∀𝑧 ∈ (ℵ‘𝐴)∃𝑤 ∈ (cf‘(ℵ‘𝐴))𝑧 ⊆ (𝑓‘𝑤))) → (∪ ran 𝑓 = (ℵ‘𝐴) ↔ ∀𝑧 ∈ (ℵ‘𝐴)∃𝑦 ∈ ran 𝑓 𝑧 ⊆ 𝑦))
65	58, 64	mpbird 256	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ On ∧ (𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) ∧ ∀𝑧 ∈ (ℵ‘𝐴)∃𝑤 ∈ (cf‘(ℵ‘𝐴))𝑧 ⊆ (𝑓‘𝑤))) → ∪ ran 𝑓 = (ℵ‘𝐴))
66	49, 65	eqtr3d 2778	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ On ∧ (𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) ∧ ∀𝑧 ∈ (ℵ‘𝐴)∃𝑤 ∈ (cf‘(ℵ‘𝐴))𝑧 ⊆ (𝑓‘𝑤))) → ∪ 𝑥 ∈ (cf‘(ℵ‘𝐴))(𝑓‘𝑥) = (ℵ‘𝐴))
67		ffvelcdm 7032	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) ∧ 𝑥 ∈ (cf‘(ℵ‘𝐴))) → (𝑓‘𝑥) ∈ (ℵ‘𝐴))
68	15	oneli 6431	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑓‘𝑥) ∈ (ℵ‘𝐴) → (𝑓‘𝑥) ∈ On)
69		harcard 9914	. . . . . . . . . . . . . . . . . . 19 ⊢ (card‘(har‘(𝑓‘𝑥))) = (har‘(𝑓‘𝑥))
70		iscard 9911	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((card‘(har‘(𝑓‘𝑥))) = (har‘(𝑓‘𝑥)) ↔ ((har‘(𝑓‘𝑥)) ∈ On ∧ ∀𝑦 ∈ (har‘(𝑓‘𝑥))𝑦 ≺ (har‘(𝑓‘𝑥))))
71	70	simprbi 497	. . . . . . . . . . . . . . . . . . 19 ⊢ ((card‘(har‘(𝑓‘𝑥))) = (har‘(𝑓‘𝑥)) → ∀𝑦 ∈ (har‘(𝑓‘𝑥))𝑦 ≺ (har‘(𝑓‘𝑥)))
72	69, 71	ax-mp 5	. . . . . . . . . . . . . . . . . 18 ⊢ ∀𝑦 ∈ (har‘(𝑓‘𝑥))𝑦 ≺ (har‘(𝑓‘𝑥))
73		domrefg 8927	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑓‘𝑥) ∈ V → (𝑓‘𝑥) ≼ (𝑓‘𝑥))
74	46, 73	ax-mp 5	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑓‘𝑥) ≼ (𝑓‘𝑥)
75		elharval 9497	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑓‘𝑥) ∈ (har‘(𝑓‘𝑥)) ↔ ((𝑓‘𝑥) ∈ On ∧ (𝑓‘𝑥) ≼ (𝑓‘𝑥)))
76	75	biimpri 227	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑓‘𝑥) ∈ On ∧ (𝑓‘𝑥) ≼ (𝑓‘𝑥)) → (𝑓‘𝑥) ∈ (har‘(𝑓‘𝑥)))
77	74, 76	mpan2 689	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑓‘𝑥) ∈ On → (𝑓‘𝑥) ∈ (har‘(𝑓‘𝑥)))
78		breq1 5108	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = (𝑓‘𝑥) → (𝑦 ≺ (har‘(𝑓‘𝑥)) ↔ (𝑓‘𝑥) ≺ (har‘(𝑓‘𝑥))))
79	78	rspccv 3578	. . . . . . . . . . . . . . . . . 18 ⊢ (∀𝑦 ∈ (har‘(𝑓‘𝑥))𝑦 ≺ (har‘(𝑓‘𝑥)) → ((𝑓‘𝑥) ∈ (har‘(𝑓‘𝑥)) → (𝑓‘𝑥) ≺ (har‘(𝑓‘𝑥))))
80	72, 77, 79	mpsyl 68	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑓‘𝑥) ∈ On → (𝑓‘𝑥) ≺ (har‘(𝑓‘𝑥)))
81	67, 68, 80	3syl 18	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) ∧ 𝑥 ∈ (cf‘(ℵ‘𝐴))) → (𝑓‘𝑥) ≺ (har‘(𝑓‘𝑥)))
82		harcl 9495	. . . . . . . . . . . . . . . . . . 19 ⊢ (har‘(𝑓‘𝑥)) ∈ On
83		2fveq3 6847	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = 𝑥 → (har‘(𝑓‘𝑦)) = (har‘(𝑓‘𝑥)))
84		pwcfsdom.1	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝐻 = (𝑦 ∈ (cf‘(ℵ‘𝐴)) ↦ (har‘(𝑓‘𝑦)))
85	83, 84	fvmptg 6946	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ∈ (cf‘(ℵ‘𝐴)) ∧ (har‘(𝑓‘𝑥)) ∈ On) → (𝐻‘𝑥) = (har‘(𝑓‘𝑥)))
86	82, 85	mpan2 689	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ (cf‘(ℵ‘𝐴)) → (𝐻‘𝑥) = (har‘(𝑓‘𝑥)))
87	86	breq2d 5117	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ (cf‘(ℵ‘𝐴)) → ((𝑓‘𝑥) ≺ (𝐻‘𝑥) ↔ (𝑓‘𝑥) ≺ (har‘(𝑓‘𝑥))))
88	87	adantl 482	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) ∧ 𝑥 ∈ (cf‘(ℵ‘𝐴))) → ((𝑓‘𝑥) ≺ (𝐻‘𝑥) ↔ (𝑓‘𝑥) ≺ (har‘(𝑓‘𝑥))))
89	81, 88	mpbird 256	. . . . . . . . . . . . . . 15 ⊢ ((𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) ∧ 𝑥 ∈ (cf‘(ℵ‘𝐴))) → (𝑓‘𝑥) ≺ (𝐻‘𝑥))
90	89	ralrimiva 3143	. . . . . . . . . . . . . 14 ⊢ (𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) → ∀𝑥 ∈ (cf‘(ℵ‘𝐴))(𝑓‘𝑥) ≺ (𝐻‘𝑥))
91		fvex 6855	. . . . . . . . . . . . . . 15 ⊢ (cf‘(ℵ‘𝐴)) ∈ V
92		eqid 2736	. . . . . . . . . . . . . . 15 ⊢ ∪ 𝑥 ∈ (cf‘(ℵ‘𝐴))(𝑓‘𝑥) = ∪ 𝑥 ∈ (cf‘(ℵ‘𝐴))(𝑓‘𝑥)
93		eqid 2736	. . . . . . . . . . . . . . 15 ⊢ X𝑥 ∈ (cf‘(ℵ‘𝐴))(𝐻‘𝑥) = X𝑥 ∈ (cf‘(ℵ‘𝐴))(𝐻‘𝑥)
94	91, 92, 93	konigth 10505	. . . . . . . . . . . . . 14 ⊢ (∀𝑥 ∈ (cf‘(ℵ‘𝐴))(𝑓‘𝑥) ≺ (𝐻‘𝑥) → ∪ 𝑥 ∈ (cf‘(ℵ‘𝐴))(𝑓‘𝑥) ≺ X𝑥 ∈ (cf‘(ℵ‘𝐴))(𝐻‘𝑥))
95	90, 94	syl 17	. . . . . . . . . . . . 13 ⊢ (𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) → ∪ 𝑥 ∈ (cf‘(ℵ‘𝐴))(𝑓‘𝑥) ≺ X𝑥 ∈ (cf‘(ℵ‘𝐴))(𝐻‘𝑥))
96	95	ad2antrl 726	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ On ∧ (𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) ∧ ∀𝑧 ∈ (ℵ‘𝐴)∃𝑤 ∈ (cf‘(ℵ‘𝐴))𝑧 ⊆ (𝑓‘𝑤))) → ∪ 𝑥 ∈ (cf‘(ℵ‘𝐴))(𝑓‘𝑥) ≺ X𝑥 ∈ (cf‘(ℵ‘𝐴))(𝐻‘𝑥))
97	66, 96	eqbrtrrd 5129	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ On ∧ (𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) ∧ ∀𝑧 ∈ (ℵ‘𝐴)∃𝑤 ∈ (cf‘(ℵ‘𝐴))𝑧 ⊆ (𝑓‘𝑤))) → (ℵ‘𝐴) ≺ X𝑥 ∈ (cf‘(ℵ‘𝐴))(𝐻‘𝑥))
98	41, 97	sylan 580	. . . . . . . . . 10 ⊢ (((𝐴 ∈ V ∧ Lim 𝐴) ∧ (𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) ∧ ∀𝑧 ∈ (ℵ‘𝐴)∃𝑤 ∈ (cf‘(ℵ‘𝐴))𝑧 ⊆ (𝑓‘𝑤))) → (ℵ‘𝐴) ≺ X𝑥 ∈ (cf‘(ℵ‘𝐴))(𝐻‘𝑥))
99		ovex 7390	. . . . . . . . . . . 12 ⊢ ((ℵ‘𝐴) ↑_m (cf‘(ℵ‘𝐴))) ∈ V
100	67	ex 413	. . . . . . . . . . . . . . . 16 ⊢ (𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) → (𝑥 ∈ (cf‘(ℵ‘𝐴)) → (𝑓‘𝑥) ∈ (ℵ‘𝐴)))
101		alephlim 10003	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ V ∧ Lim 𝐴) → (ℵ‘𝐴) = ∪ 𝑦 ∈ 𝐴 (ℵ‘𝑦))
102	101	eleq2d 2823	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ V ∧ Lim 𝐴) → ((𝑓‘𝑥) ∈ (ℵ‘𝐴) ↔ (𝑓‘𝑥) ∈ ∪ 𝑦 ∈ 𝐴 (ℵ‘𝑦)))
103		eliun 4958	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓‘𝑥) ∈ ∪ 𝑦 ∈ 𝐴 (ℵ‘𝑦) ↔ ∃𝑦 ∈ 𝐴 (𝑓‘𝑥) ∈ (ℵ‘𝑦))
104		alephcard 10006	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (card‘(ℵ‘𝑦)) = (ℵ‘𝑦)
105	104	eleq2i 2829	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑓‘𝑥) ∈ (card‘(ℵ‘𝑦)) ↔ (𝑓‘𝑥) ∈ (ℵ‘𝑦))
106		cardsdomelir 9909	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑓‘𝑥) ∈ (card‘(ℵ‘𝑦)) → (𝑓‘𝑥) ≺ (ℵ‘𝑦))
107	105, 106	sylbir 234	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑓‘𝑥) ∈ (ℵ‘𝑦) → (𝑓‘𝑥) ≺ (ℵ‘𝑦))
108		elharval 9497	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((ℵ‘𝑦) ∈ (har‘(𝑓‘𝑥)) ↔ ((ℵ‘𝑦) ∈ On ∧ (ℵ‘𝑦) ≼ (𝑓‘𝑥)))
109	108	simprbi 497	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((ℵ‘𝑦) ∈ (har‘(𝑓‘𝑥)) → (ℵ‘𝑦) ≼ (𝑓‘𝑥))
110		domnsym 9043	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((ℵ‘𝑦) ≼ (𝑓‘𝑥) → ¬ (𝑓‘𝑥) ≺ (ℵ‘𝑦))
111	109, 110	syl 17	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((ℵ‘𝑦) ∈ (har‘(𝑓‘𝑥)) → ¬ (𝑓‘𝑥) ≺ (ℵ‘𝑦))
112	111	con2i 139	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑓‘𝑥) ≺ (ℵ‘𝑦) → ¬ (ℵ‘𝑦) ∈ (har‘(𝑓‘𝑥)))
113		alephon 10005	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (ℵ‘𝑦) ∈ On
114		ontri1 6351	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((har‘(𝑓‘𝑥)) ∈ On ∧ (ℵ‘𝑦) ∈ On) → ((har‘(𝑓‘𝑥)) ⊆ (ℵ‘𝑦) ↔ ¬ (ℵ‘𝑦) ∈ (har‘(𝑓‘𝑥))))
115	82, 113, 114	mp2an 690	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((har‘(𝑓‘𝑥)) ⊆ (ℵ‘𝑦) ↔ ¬ (ℵ‘𝑦) ∈ (har‘(𝑓‘𝑥)))
116	112, 115	sylibr 233	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑓‘𝑥) ≺ (ℵ‘𝑦) → (har‘(𝑓‘𝑥)) ⊆ (ℵ‘𝑦))
117	107, 116	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑓‘𝑥) ∈ (ℵ‘𝑦) → (har‘(𝑓‘𝑥)) ⊆ (ℵ‘𝑦))
118		alephord2i 10013	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐴 ∈ On → (𝑦 ∈ 𝐴 → (ℵ‘𝑦) ∈ (ℵ‘𝐴)))
119	118	imp 407	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ 𝐴) → (ℵ‘𝑦) ∈ (ℵ‘𝐴))
120		ontr2 6364	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((har‘(𝑓‘𝑥)) ∈ On ∧ (ℵ‘𝐴) ∈ On) → (((har‘(𝑓‘𝑥)) ⊆ (ℵ‘𝑦) ∧ (ℵ‘𝑦) ∈ (ℵ‘𝐴)) → (har‘(𝑓‘𝑥)) ∈ (ℵ‘𝐴)))
121	82, 15, 120	mp2an 690	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((har‘(𝑓‘𝑥)) ⊆ (ℵ‘𝑦) ∧ (ℵ‘𝑦) ∈ (ℵ‘𝐴)) → (har‘(𝑓‘𝑥)) ∈ (ℵ‘𝐴))
122	117, 119, 121	syl2anr 597	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐴 ∈ On ∧ 𝑦 ∈ 𝐴) ∧ (𝑓‘𝑥) ∈ (ℵ‘𝑦)) → (har‘(𝑓‘𝑥)) ∈ (ℵ‘𝐴))
123	122	rexlimdva2 3154	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐴 ∈ On → (∃𝑦 ∈ 𝐴 (𝑓‘𝑥) ∈ (ℵ‘𝑦) → (har‘(𝑓‘𝑥)) ∈ (ℵ‘𝐴)))
124	103, 123	biimtrid 241	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 ∈ On → ((𝑓‘𝑥) ∈ ∪ 𝑦 ∈ 𝐴 (ℵ‘𝑦) → (har‘(𝑓‘𝑥)) ∈ (ℵ‘𝐴)))
125	41, 124	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ V ∧ Lim 𝐴) → ((𝑓‘𝑥) ∈ ∪ 𝑦 ∈ 𝐴 (ℵ‘𝑦) → (har‘(𝑓‘𝑥)) ∈ (ℵ‘𝐴)))
126	102, 125	sylbid 239	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ V ∧ Lim 𝐴) → ((𝑓‘𝑥) ∈ (ℵ‘𝐴) → (har‘(𝑓‘𝑥)) ∈ (ℵ‘𝐴)))
127	100, 126	sylan9r 509	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ V ∧ Lim 𝐴) ∧ 𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴)) → (𝑥 ∈ (cf‘(ℵ‘𝐴)) → (har‘(𝑓‘𝑥)) ∈ (ℵ‘𝐴)))
128	127	imp 407	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ V ∧ Lim 𝐴) ∧ 𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴)) ∧ 𝑥 ∈ (cf‘(ℵ‘𝐴))) → (har‘(𝑓‘𝑥)) ∈ (ℵ‘𝐴))
129	83	cbvmptv 5218	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ (cf‘(ℵ‘𝐴)) ↦ (har‘(𝑓‘𝑦))) = (𝑥 ∈ (cf‘(ℵ‘𝐴)) ↦ (har‘(𝑓‘𝑥)))
130	84, 129	eqtri 2764	. . . . . . . . . . . . . 14 ⊢ 𝐻 = (𝑥 ∈ (cf‘(ℵ‘𝐴)) ↦ (har‘(𝑓‘𝑥)))
131	128, 130	fmptd 7062	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ V ∧ Lim 𝐴) ∧ 𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴)) → 𝐻:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴))
132		ffvelcdm 7032	. . . . . . . . . . . . . . 15 ⊢ ((𝐻:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) ∧ 𝑥 ∈ (cf‘(ℵ‘𝐴))) → (𝐻‘𝑥) ∈ (ℵ‘𝐴))
133		onelss 6359	. . . . . . . . . . . . . . 15 ⊢ ((ℵ‘𝐴) ∈ On → ((𝐻‘𝑥) ∈ (ℵ‘𝐴) → (𝐻‘𝑥) ⊆ (ℵ‘𝐴)))
134	15, 132, 133	mpsyl 68	. . . . . . . . . . . . . 14 ⊢ ((𝐻:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) ∧ 𝑥 ∈ (cf‘(ℵ‘𝐴))) → (𝐻‘𝑥) ⊆ (ℵ‘𝐴))
135	134	ralrimiva 3143	. . . . . . . . . . . . 13 ⊢ (𝐻:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) → ∀𝑥 ∈ (cf‘(ℵ‘𝐴))(𝐻‘𝑥) ⊆ (ℵ‘𝐴))
136		ss2ixp 8848	. . . . . . . . . . . . . 14 ⊢ (∀𝑥 ∈ (cf‘(ℵ‘𝐴))(𝐻‘𝑥) ⊆ (ℵ‘𝐴) → X𝑥 ∈ (cf‘(ℵ‘𝐴))(𝐻‘𝑥) ⊆ X𝑥 ∈ (cf‘(ℵ‘𝐴))(ℵ‘𝐴))
137	91, 10	ixpconst 8845	. . . . . . . . . . . . . 14 ⊢ X𝑥 ∈ (cf‘(ℵ‘𝐴))(ℵ‘𝐴) = ((ℵ‘𝐴) ↑_m (cf‘(ℵ‘𝐴)))
138	136, 137	sseqtrdi 3994	. . . . . . . . . . . . 13 ⊢ (∀𝑥 ∈ (cf‘(ℵ‘𝐴))(𝐻‘𝑥) ⊆ (ℵ‘𝐴) → X𝑥 ∈ (cf‘(ℵ‘𝐴))(𝐻‘𝑥) ⊆ ((ℵ‘𝐴) ↑_m (cf‘(ℵ‘𝐴))))
139	131, 135, 138	3syl 18	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ V ∧ Lim 𝐴) ∧ 𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴)) → X𝑥 ∈ (cf‘(ℵ‘𝐴))(𝐻‘𝑥) ⊆ ((ℵ‘𝐴) ↑_m (cf‘(ℵ‘𝐴))))
140		ssdomg 8940	. . . . . . . . . . . 12 ⊢ (((ℵ‘𝐴) ↑_m (cf‘(ℵ‘𝐴))) ∈ V → (X𝑥 ∈ (cf‘(ℵ‘𝐴))(𝐻‘𝑥) ⊆ ((ℵ‘𝐴) ↑_m (cf‘(ℵ‘𝐴))) → X𝑥 ∈ (cf‘(ℵ‘𝐴))(𝐻‘𝑥) ≼ ((ℵ‘𝐴) ↑_m (cf‘(ℵ‘𝐴)))))
141	99, 139, 140	mpsyl 68	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ V ∧ Lim 𝐴) ∧ 𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴)) → X𝑥 ∈ (cf‘(ℵ‘𝐴))(𝐻‘𝑥) ≼ ((ℵ‘𝐴) ↑_m (cf‘(ℵ‘𝐴))))
142	141	adantrr 715	. . . . . . . . . 10 ⊢ (((𝐴 ∈ V ∧ Lim 𝐴) ∧ (𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) ∧ ∀𝑧 ∈ (ℵ‘𝐴)∃𝑤 ∈ (cf‘(ℵ‘𝐴))𝑧 ⊆ (𝑓‘𝑤))) → X𝑥 ∈ (cf‘(ℵ‘𝐴))(𝐻‘𝑥) ≼ ((ℵ‘𝐴) ↑_m (cf‘(ℵ‘𝐴))))
143		sdomdomtr 9054	. . . . . . . . . 10 ⊢ (((ℵ‘𝐴) ≺ X𝑥 ∈ (cf‘(ℵ‘𝐴))(𝐻‘𝑥) ∧ X𝑥 ∈ (cf‘(ℵ‘𝐴))(𝐻‘𝑥) ≼ ((ℵ‘𝐴) ↑_m (cf‘(ℵ‘𝐴)))) → (ℵ‘𝐴) ≺ ((ℵ‘𝐴) ↑_m (cf‘(ℵ‘𝐴))))
144	98, 142, 143	syl2anc 584	. . . . . . . . 9 ⊢ (((𝐴 ∈ V ∧ Lim 𝐴) ∧ (𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) ∧ ∀𝑧 ∈ (ℵ‘𝐴)∃𝑤 ∈ (cf‘(ℵ‘𝐴))𝑧 ⊆ (𝑓‘𝑤))) → (ℵ‘𝐴) ≺ ((ℵ‘𝐴) ↑_m (cf‘(ℵ‘𝐴))))
145	144	expcom 414	. . . . . . . 8 ⊢ ((𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) ∧ ∀𝑧 ∈ (ℵ‘𝐴)∃𝑤 ∈ (cf‘(ℵ‘𝐴))𝑧 ⊆ (𝑓‘𝑤)) → ((𝐴 ∈ V ∧ Lim 𝐴) → (ℵ‘𝐴) ≺ ((ℵ‘𝐴) ↑_m (cf‘(ℵ‘𝐴)))))
146	145	3adant2 1131	. . . . . . 7 ⊢ ((𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) ∧ Smo 𝑓 ∧ ∀𝑧 ∈ (ℵ‘𝐴)∃𝑤 ∈ (cf‘(ℵ‘𝐴))𝑧 ⊆ (𝑓‘𝑤)) → ((𝐴 ∈ V ∧ Lim 𝐴) → (ℵ‘𝐴) ≺ ((ℵ‘𝐴) ↑_m (cf‘(ℵ‘𝐴)))))
147	146	exlimiv 1933	. . . . . 6 ⊢ (∃𝑓(𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) ∧ Smo 𝑓 ∧ ∀𝑧 ∈ (ℵ‘𝐴)∃𝑤 ∈ (cf‘(ℵ‘𝐴))𝑧 ⊆ (𝑓‘𝑤)) → ((𝐴 ∈ V ∧ Lim 𝐴) → (ℵ‘𝐴) ≺ ((ℵ‘𝐴) ↑_m (cf‘(ℵ‘𝐴)))))
148	15, 40, 147	mp2b 10	. . . . 5 ⊢ ((𝐴 ∈ V ∧ Lim 𝐴) → (ℵ‘𝐴) ≺ ((ℵ‘𝐴) ↑_m (cf‘(ℵ‘𝐴))))
149	148	a1i 11	. . . 4 ⊢ (𝐴 ∈ On → ((𝐴 ∈ V ∧ Lim 𝐴) → (ℵ‘𝐴) ≺ ((ℵ‘𝐴) ↑_m (cf‘(ℵ‘𝐴)))))
150	33, 39, 149	3jaod 1428	. . 3 ⊢ (𝐴 ∈ On → ((𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ (𝐴 ∈ V ∧ Lim 𝐴)) → (ℵ‘𝐴) ≺ ((ℵ‘𝐴) ↑_m (cf‘(ℵ‘𝐴)))))
151	2, 150	mpd 15	. 2 ⊢ (𝐴 ∈ On → (ℵ‘𝐴) ≺ ((ℵ‘𝐴) ↑_m (cf‘(ℵ‘𝐴))))
152		alephfnon 10001	. . . . 5 ⊢ ℵ Fn On
153	152	fndmi 6606	. . . 4 ⊢ dom ℵ = On
154	153	eleq2i 2829	. . 3 ⊢ (𝐴 ∈ dom ℵ ↔ 𝐴 ∈ On)
155		ndmfv 6877	. . . 4 ⊢ (¬ 𝐴 ∈ dom ℵ → (ℵ‘𝐴) = ∅)
156		1n0 8434	. . . . . 6 ⊢ 1_o ≠ ∅
157		1oex 8422	. . . . . . 7 ⊢ 1_o ∈ V
158	157	0sdom 9051	. . . . . 6 ⊢ (∅ ≺ 1_o ↔ 1_o ≠ ∅)
159	156, 158	mpbir 230	. . . . 5 ⊢ ∅ ≺ 1_o
160		id 22	. . . . . 6 ⊢ ((ℵ‘𝐴) = ∅ → (ℵ‘𝐴) = ∅)
161		fveq2 6842	. . . . . . . . 9 ⊢ ((ℵ‘𝐴) = ∅ → (cf‘(ℵ‘𝐴)) = (cf‘∅))
162		cf0 10187	. . . . . . . . 9 ⊢ (cf‘∅) = ∅
163	161, 162	eqtrdi 2792	. . . . . . . 8 ⊢ ((ℵ‘𝐴) = ∅ → (cf‘(ℵ‘𝐴)) = ∅)
164	160, 163	oveq12d 7375	. . . . . . 7 ⊢ ((ℵ‘𝐴) = ∅ → ((ℵ‘𝐴) ↑_m (cf‘(ℵ‘𝐴))) = (∅ ↑_m ∅))
165		0ex 5264	. . . . . . . 8 ⊢ ∅ ∈ V
166		map0e 8820	. . . . . . . 8 ⊢ (∅ ∈ V → (∅ ↑_m ∅) = 1_o)
167	165, 166	ax-mp 5	. . . . . . 7 ⊢ (∅ ↑_m ∅) = 1_o
168	164, 167	eqtrdi 2792	. . . . . 6 ⊢ ((ℵ‘𝐴) = ∅ → ((ℵ‘𝐴) ↑_m (cf‘(ℵ‘𝐴))) = 1_o)
169	160, 168	breq12d 5118	. . . . 5 ⊢ ((ℵ‘𝐴) = ∅ → ((ℵ‘𝐴) ≺ ((ℵ‘𝐴) ↑_m (cf‘(ℵ‘𝐴))) ↔ ∅ ≺ 1_o))
170	159, 169	mpbiri 257	. . . 4 ⊢ ((ℵ‘𝐴) = ∅ → (ℵ‘𝐴) ≺ ((ℵ‘𝐴) ↑_m (cf‘(ℵ‘𝐴))))
171	155, 170	syl 17	. . 3 ⊢ (¬ 𝐴 ∈ dom ℵ → (ℵ‘𝐴) ≺ ((ℵ‘𝐴) ↑_m (cf‘(ℵ‘𝐴))))
172	154, 171	sylnbir 330	. 2 ⊢ (¬ 𝐴 ∈ On → (ℵ‘𝐴) ≺ ((ℵ‘𝐴) ↑_m (cf‘(ℵ‘𝐴))))
173	151, 172	pm2.61i 182	1 ⊢ (ℵ‘𝐴) ≺ ((ℵ‘𝐴) ↑_m (cf‘(ℵ‘𝐴)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 ∨ w3o 1086 ∧ w3a 1087 = wceq 1541 ∃wex 1781 ∈ wcel 2106 {cab 2713 ≠ wne 2943 ∀wral 3064 ∃wrex 3073 Vcvv 3445 ⊆ wss 3910 ∅c0 4282 𝒫 cpw 4560 ∪ cuni 4865 ∪ ciun 4954 class class class wbr 5105 ↦ cmpt 5188 dom cdm 5633 ran crn 5634 Oncon0 6317 Lim wlim 6318 suc csuc 6319 Fn wfn 6491 ⟶wf 6492 ‘cfv 6496 (class class class)co 7357 ωcom 7802 Smo wsmo 8291 1_oc1o 8405 2_oc2o 8406 ↑_m cmap 8765 Xcixp 8835 ≈ cen 8880 ≼ cdom 8881 ≺ csdm 8882 harchar 9492 cardccrd 9871 ℵcale 9872 cfccf 9873

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-inf2 9577 ax-ac2 10399

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-iin 4957 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-se 5589 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-isom 6505 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-om 7803 df-1st 7921 df-2nd 7922 df-frecs 8212 df-wrecs 8243 df-smo 8292 df-recs 8317 df-rdg 8356 df-1o 8412 df-2o 8413 df-er 8648 df-map 8767 df-ixp 8836 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-oi 9446 df-har 9493 df-card 9875 df-aleph 9876 df-cf 9877 df-acn 9878 df-ac 10052

This theorem is referenced by: cfpwsdom 10520 tskcard 10717 bj-pwcfsdom 35533

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