Theorem cfslb2n 10287

Description: Any small collection of small subsets of 𝐴 cannot have union 𝐴, where "small" means smaller than the cofinality. This is a stronger version of cfslb 10285. This is a common application of cofinality: under AC, (ℵ‘1) is regular, so it is not a countable union of countable sets. (Contributed by Mario Carneiro, 24-Jun-2013.)

Hypothesis

Ref	Expression
cfslb.1	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
cfslb2n	⊢ ((Lim 𝐴 ∧ ∀𝑥 ∈ 𝐵 (𝑥 ⊆ 𝐴 ∧ 𝑥 ≺ (cf‘𝐴))) → (𝐵 ≺ (cf‘𝐴) → ∪ 𝐵 ≠ 𝐴))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem cfslb2n

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		limord 6418	. . . . . . . . . 10 ⊢ (Lim 𝐴 → Ord 𝐴)
2		ordsson 7782	. . . . . . . . . 10 ⊢ (Ord 𝐴 → 𝐴 ⊆ On)
3		sstr 3972	. . . . . . . . . . 11 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝐴 ⊆ On) → 𝑥 ⊆ On)
4	3	expcom 413	. . . . . . . . . 10 ⊢ (𝐴 ⊆ On → (𝑥 ⊆ 𝐴 → 𝑥 ⊆ On))
5	1, 2, 4	3syl 18	. . . . . . . . 9 ⊢ (Lim 𝐴 → (𝑥 ⊆ 𝐴 → 𝑥 ⊆ On))
6		onsucuni 7827	. . . . . . . . 9 ⊢ (𝑥 ⊆ On → 𝑥 ⊆ suc ∪ 𝑥)
7	5, 6	syl6 35	. . . . . . . 8 ⊢ (Lim 𝐴 → (𝑥 ⊆ 𝐴 → 𝑥 ⊆ suc ∪ 𝑥))
8	7	adantrd 491	. . . . . . 7 ⊢ (Lim 𝐴 → ((𝑥 ⊆ 𝐴 ∧ 𝑥 ≺ (cf‘𝐴)) → 𝑥 ⊆ suc ∪ 𝑥))
9	8	ralimdv 3155	. . . . . 6 ⊢ (Lim 𝐴 → (∀𝑥 ∈ 𝐵 (𝑥 ⊆ 𝐴 ∧ 𝑥 ≺ (cf‘𝐴)) → ∀𝑥 ∈ 𝐵 𝑥 ⊆ suc ∪ 𝑥))
10		uniiun 5039	. . . . . . 7 ⊢ ∪ 𝐵 = ∪ 𝑥 ∈ 𝐵 𝑥
11		ss2iun 4991	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐵 𝑥 ⊆ suc ∪ 𝑥 → ∪ 𝑥 ∈ 𝐵 𝑥 ⊆ ∪ 𝑥 ∈ 𝐵 suc ∪ 𝑥)
12	10, 11	eqsstrid 4002	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐵 𝑥 ⊆ suc ∪ 𝑥 → ∪ 𝐵 ⊆ ∪ 𝑥 ∈ 𝐵 suc ∪ 𝑥)
13	9, 12	syl6 35	. . . . 5 ⊢ (Lim 𝐴 → (∀𝑥 ∈ 𝐵 (𝑥 ⊆ 𝐴 ∧ 𝑥 ≺ (cf‘𝐴)) → ∪ 𝐵 ⊆ ∪ 𝑥 ∈ 𝐵 suc ∪ 𝑥))
14	13	imp 406	. . . 4 ⊢ ((Lim 𝐴 ∧ ∀𝑥 ∈ 𝐵 (𝑥 ⊆ 𝐴 ∧ 𝑥 ≺ (cf‘𝐴))) → ∪ 𝐵 ⊆ ∪ 𝑥 ∈ 𝐵 suc ∪ 𝑥)
15		cfslb.1	. . . . . . . . . 10 ⊢ 𝐴 ∈ V
16	15	cfslbn 10286	. . . . . . . . 9 ⊢ ((Lim 𝐴 ∧ 𝑥 ⊆ 𝐴 ∧ 𝑥 ≺ (cf‘𝐴)) → ∪ 𝑥 ∈ 𝐴)
17	16	3expib 1122	. . . . . . . 8 ⊢ (Lim 𝐴 → ((𝑥 ⊆ 𝐴 ∧ 𝑥 ≺ (cf‘𝐴)) → ∪ 𝑥 ∈ 𝐴))
18		ordsucss 7817	. . . . . . . 8 ⊢ (Ord 𝐴 → (∪ 𝑥 ∈ 𝐴 → suc ∪ 𝑥 ⊆ 𝐴))
19	1, 17, 18	sylsyld 61	. . . . . . 7 ⊢ (Lim 𝐴 → ((𝑥 ⊆ 𝐴 ∧ 𝑥 ≺ (cf‘𝐴)) → suc ∪ 𝑥 ⊆ 𝐴))
20	19	ralimdv 3155	. . . . . 6 ⊢ (Lim 𝐴 → (∀𝑥 ∈ 𝐵 (𝑥 ⊆ 𝐴 ∧ 𝑥 ≺ (cf‘𝐴)) → ∀𝑥 ∈ 𝐵 suc ∪ 𝑥 ⊆ 𝐴))
21		iunss 5026	. . . . . 6 ⊢ (∪ 𝑥 ∈ 𝐵 suc ∪ 𝑥 ⊆ 𝐴 ↔ ∀𝑥 ∈ 𝐵 suc ∪ 𝑥 ⊆ 𝐴)
22	20, 21	imbitrrdi 252	. . . . 5 ⊢ (Lim 𝐴 → (∀𝑥 ∈ 𝐵 (𝑥 ⊆ 𝐴 ∧ 𝑥 ≺ (cf‘𝐴)) → ∪ 𝑥 ∈ 𝐵 suc ∪ 𝑥 ⊆ 𝐴))
23	22	imp 406	. . . 4 ⊢ ((Lim 𝐴 ∧ ∀𝑥 ∈ 𝐵 (𝑥 ⊆ 𝐴 ∧ 𝑥 ≺ (cf‘𝐴))) → ∪ 𝑥 ∈ 𝐵 suc ∪ 𝑥 ⊆ 𝐴)
24		sseq1 3989	. . . . . 6 ⊢ (∪ 𝐵 = 𝐴 → (∪ 𝐵 ⊆ ∪ 𝑥 ∈ 𝐵 suc ∪ 𝑥 ↔ 𝐴 ⊆ ∪ 𝑥 ∈ 𝐵 suc ∪ 𝑥))
25		eqss 3979	. . . . . . 7 ⊢ (∪ 𝑥 ∈ 𝐵 suc ∪ 𝑥 = 𝐴 ↔ (∪ 𝑥 ∈ 𝐵 suc ∪ 𝑥 ⊆ 𝐴 ∧ 𝐴 ⊆ ∪ 𝑥 ∈ 𝐵 suc ∪ 𝑥))
26	25	simplbi2com 502	. . . . . 6 ⊢ (𝐴 ⊆ ∪ 𝑥 ∈ 𝐵 suc ∪ 𝑥 → (∪ 𝑥 ∈ 𝐵 suc ∪ 𝑥 ⊆ 𝐴 → ∪ 𝑥 ∈ 𝐵 suc ∪ 𝑥 = 𝐴))
27	24, 26	biimtrdi 253	. . . . 5 ⊢ (∪ 𝐵 = 𝐴 → (∪ 𝐵 ⊆ ∪ 𝑥 ∈ 𝐵 suc ∪ 𝑥 → (∪ 𝑥 ∈ 𝐵 suc ∪ 𝑥 ⊆ 𝐴 → ∪ 𝑥 ∈ 𝐵 suc ∪ 𝑥 = 𝐴)))
28	27	com3l 89	. . . 4 ⊢ (∪ 𝐵 ⊆ ∪ 𝑥 ∈ 𝐵 suc ∪ 𝑥 → (∪ 𝑥 ∈ 𝐵 suc ∪ 𝑥 ⊆ 𝐴 → (∪ 𝐵 = 𝐴 → ∪ 𝑥 ∈ 𝐵 suc ∪ 𝑥 = 𝐴)))
29	14, 23, 28	sylc 65	. . 3 ⊢ ((Lim 𝐴 ∧ ∀𝑥 ∈ 𝐵 (𝑥 ⊆ 𝐴 ∧ 𝑥 ≺ (cf‘𝐴))) → (∪ 𝐵 = 𝐴 → ∪ 𝑥 ∈ 𝐵 suc ∪ 𝑥 = 𝐴))
30		limsuc 7849	. . . . . . . . 9 ⊢ (Lim 𝐴 → (∪ 𝑥 ∈ 𝐴 ↔ suc ∪ 𝑥 ∈ 𝐴))
31	17, 30	sylibd 239	. . . . . . . 8 ⊢ (Lim 𝐴 → ((𝑥 ⊆ 𝐴 ∧ 𝑥 ≺ (cf‘𝐴)) → suc ∪ 𝑥 ∈ 𝐴))
32	31	ralimdv 3155	. . . . . . 7 ⊢ (Lim 𝐴 → (∀𝑥 ∈ 𝐵 (𝑥 ⊆ 𝐴 ∧ 𝑥 ≺ (cf‘𝐴)) → ∀𝑥 ∈ 𝐵 suc ∪ 𝑥 ∈ 𝐴))
33	32	imp 406	. . . . . 6 ⊢ ((Lim 𝐴 ∧ ∀𝑥 ∈ 𝐵 (𝑥 ⊆ 𝐴 ∧ 𝑥 ≺ (cf‘𝐴))) → ∀𝑥 ∈ 𝐵 suc ∪ 𝑥 ∈ 𝐴)
34		r19.29 3102	. . . . . . . 8 ⊢ ((∀𝑥 ∈ 𝐵 suc ∪ 𝑥 ∈ 𝐴 ∧ ∃𝑥 ∈ 𝐵 𝑦 = suc ∪ 𝑥) → ∃𝑥 ∈ 𝐵 (suc ∪ 𝑥 ∈ 𝐴 ∧ 𝑦 = suc ∪ 𝑥))
35		eleq1 2823	. . . . . . . . . 10 ⊢ (𝑦 = suc ∪ 𝑥 → (𝑦 ∈ 𝐴 ↔ suc ∪ 𝑥 ∈ 𝐴))
36	35	biimparc 479	. . . . . . . . 9 ⊢ ((suc ∪ 𝑥 ∈ 𝐴 ∧ 𝑦 = suc ∪ 𝑥) → 𝑦 ∈ 𝐴)
37	36	rexlimivw 3138	. . . . . . . 8 ⊢ (∃𝑥 ∈ 𝐵 (suc ∪ 𝑥 ∈ 𝐴 ∧ 𝑦 = suc ∪ 𝑥) → 𝑦 ∈ 𝐴)
38	34, 37	syl 17	. . . . . . 7 ⊢ ((∀𝑥 ∈ 𝐵 suc ∪ 𝑥 ∈ 𝐴 ∧ ∃𝑥 ∈ 𝐵 𝑦 = suc ∪ 𝑥) → 𝑦 ∈ 𝐴)
39	38	ex 412	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐵 suc ∪ 𝑥 ∈ 𝐴 → (∃𝑥 ∈ 𝐵 𝑦 = suc ∪ 𝑥 → 𝑦 ∈ 𝐴))
40	33, 39	syl 17	. . . . 5 ⊢ ((Lim 𝐴 ∧ ∀𝑥 ∈ 𝐵 (𝑥 ⊆ 𝐴 ∧ 𝑥 ≺ (cf‘𝐴))) → (∃𝑥 ∈ 𝐵 𝑦 = suc ∪ 𝑥 → 𝑦 ∈ 𝐴))
41	40	abssdv 4048	. . . 4 ⊢ ((Lim 𝐴 ∧ ∀𝑥 ∈ 𝐵 (𝑥 ⊆ 𝐴 ∧ 𝑥 ≺ (cf‘𝐴))) → {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑦 = suc ∪ 𝑥} ⊆ 𝐴)
42		vuniex 7738	. . . . . . . 8 ⊢ ∪ 𝑥 ∈ V
43	42	sucex 7805	. . . . . . 7 ⊢ suc ∪ 𝑥 ∈ V
44	43	dfiun2 5014	. . . . . 6 ⊢ ∪ 𝑥 ∈ 𝐵 suc ∪ 𝑥 = ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑦 = suc ∪ 𝑥}
45	44	eqeq1i 2741	. . . . 5 ⊢ (∪ 𝑥 ∈ 𝐵 suc ∪ 𝑥 = 𝐴 ↔ ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑦 = suc ∪ 𝑥} = 𝐴)
46	15	cfslb 10285	. . . . . 6 ⊢ ((Lim 𝐴 ∧ {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑦 = suc ∪ 𝑥} ⊆ 𝐴 ∧ ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑦 = suc ∪ 𝑥} = 𝐴) → (cf‘𝐴) ≼ {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑦 = suc ∪ 𝑥})
47	46	3expia 1121	. . . . 5 ⊢ ((Lim 𝐴 ∧ {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑦 = suc ∪ 𝑥} ⊆ 𝐴) → (∪ {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑦 = suc ∪ 𝑥} = 𝐴 → (cf‘𝐴) ≼ {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑦 = suc ∪ 𝑥}))
48	45, 47	biimtrid 242	. . . 4 ⊢ ((Lim 𝐴 ∧ {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑦 = suc ∪ 𝑥} ⊆ 𝐴) → (∪ 𝑥 ∈ 𝐵 suc ∪ 𝑥 = 𝐴 → (cf‘𝐴) ≼ {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑦 = suc ∪ 𝑥}))
49	41, 48	syldan 591	. . 3 ⊢ ((Lim 𝐴 ∧ ∀𝑥 ∈ 𝐵 (𝑥 ⊆ 𝐴 ∧ 𝑥 ≺ (cf‘𝐴))) → (∪ 𝑥 ∈ 𝐵 suc ∪ 𝑥 = 𝐴 → (cf‘𝐴) ≼ {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑦 = suc ∪ 𝑥}))
50		eqid 2736	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐵 ↦ suc ∪ 𝑥) = (𝑥 ∈ 𝐵 ↦ suc ∪ 𝑥)
51	50	rnmpt 5942	. . . . . . . 8 ⊢ ran (𝑥 ∈ 𝐵 ↦ suc ∪ 𝑥) = {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑦 = suc ∪ 𝑥}
52	43, 50	fnmpti 6686	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐵 ↦ suc ∪ 𝑥) Fn 𝐵
53		dffn4 6801	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐵 ↦ suc ∪ 𝑥) Fn 𝐵 ↔ (𝑥 ∈ 𝐵 ↦ suc ∪ 𝑥):𝐵–onto→ran (𝑥 ∈ 𝐵 ↦ suc ∪ 𝑥))
54	52, 53	mpbi 230	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐵 ↦ suc ∪ 𝑥):𝐵–onto→ran (𝑥 ∈ 𝐵 ↦ suc ∪ 𝑥)
55		relsdom 8971	. . . . . . . . . . 11 ⊢ Rel ≺
56	55	brrelex1i 5715	. . . . . . . . . 10 ⊢ (𝐵 ≺ (cf‘𝐴) → 𝐵 ∈ V)
57		breq1 5127	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝐵 → (𝑦 ≺ (cf‘𝐴) ↔ 𝐵 ≺ (cf‘𝐴)))
58		foeq2 6792	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝐵 → ((𝑥 ∈ 𝐵 ↦ suc ∪ 𝑥):𝑦–onto→ran (𝑥 ∈ 𝐵 ↦ suc ∪ 𝑥) ↔ (𝑥 ∈ 𝐵 ↦ suc ∪ 𝑥):𝐵–onto→ran (𝑥 ∈ 𝐵 ↦ suc ∪ 𝑥)))
59		breq2 5128	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝐵 → (ran (𝑥 ∈ 𝐵 ↦ suc ∪ 𝑥) ≼ 𝑦 ↔ ran (𝑥 ∈ 𝐵 ↦ suc ∪ 𝑥) ≼ 𝐵))
60	58, 59	imbi12d 344	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝐵 → (((𝑥 ∈ 𝐵 ↦ suc ∪ 𝑥):𝑦–onto→ran (𝑥 ∈ 𝐵 ↦ suc ∪ 𝑥) → ran (𝑥 ∈ 𝐵 ↦ suc ∪ 𝑥) ≼ 𝑦) ↔ ((𝑥 ∈ 𝐵 ↦ suc ∪ 𝑥):𝐵–onto→ran (𝑥 ∈ 𝐵 ↦ suc ∪ 𝑥) → ran (𝑥 ∈ 𝐵 ↦ suc ∪ 𝑥) ≼ 𝐵)))
61	57, 60	imbi12d 344	. . . . . . . . . . 11 ⊢ (𝑦 = 𝐵 → ((𝑦 ≺ (cf‘𝐴) → ((𝑥 ∈ 𝐵 ↦ suc ∪ 𝑥):𝑦–onto→ran (𝑥 ∈ 𝐵 ↦ suc ∪ 𝑥) → ran (𝑥 ∈ 𝐵 ↦ suc ∪ 𝑥) ≼ 𝑦)) ↔ (𝐵 ≺ (cf‘𝐴) → ((𝑥 ∈ 𝐵 ↦ suc ∪ 𝑥):𝐵–onto→ran (𝑥 ∈ 𝐵 ↦ suc ∪ 𝑥) → ran (𝑥 ∈ 𝐵 ↦ suc ∪ 𝑥) ≼ 𝐵))))
62		cfon 10274	. . . . . . . . . . . . 13 ⊢ (cf‘𝐴) ∈ On
63		sdomdom 8999	. . . . . . . . . . . . 13 ⊢ (𝑦 ≺ (cf‘𝐴) → 𝑦 ≼ (cf‘𝐴))
64		ondomen 10056	. . . . . . . . . . . . 13 ⊢ (((cf‘𝐴) ∈ On ∧ 𝑦 ≼ (cf‘𝐴)) → 𝑦 ∈ dom card)
65	62, 63, 64	sylancr 587	. . . . . . . . . . . 12 ⊢ (𝑦 ≺ (cf‘𝐴) → 𝑦 ∈ dom card)
66		fodomnum 10076	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ dom card → ((𝑥 ∈ 𝐵 ↦ suc ∪ 𝑥):𝑦–onto→ran (𝑥 ∈ 𝐵 ↦ suc ∪ 𝑥) → ran (𝑥 ∈ 𝐵 ↦ suc ∪ 𝑥) ≼ 𝑦))
67	65, 66	syl 17	. . . . . . . . . . 11 ⊢ (𝑦 ≺ (cf‘𝐴) → ((𝑥 ∈ 𝐵 ↦ suc ∪ 𝑥):𝑦–onto→ran (𝑥 ∈ 𝐵 ↦ suc ∪ 𝑥) → ran (𝑥 ∈ 𝐵 ↦ suc ∪ 𝑥) ≼ 𝑦))
68	61, 67	vtoclg 3538	. . . . . . . . . 10 ⊢ (𝐵 ∈ V → (𝐵 ≺ (cf‘𝐴) → ((𝑥 ∈ 𝐵 ↦ suc ∪ 𝑥):𝐵–onto→ran (𝑥 ∈ 𝐵 ↦ suc ∪ 𝑥) → ran (𝑥 ∈ 𝐵 ↦ suc ∪ 𝑥) ≼ 𝐵)))
69	56, 68	mpcom 38	. . . . . . . . 9 ⊢ (𝐵 ≺ (cf‘𝐴) → ((𝑥 ∈ 𝐵 ↦ suc ∪ 𝑥):𝐵–onto→ran (𝑥 ∈ 𝐵 ↦ suc ∪ 𝑥) → ran (𝑥 ∈ 𝐵 ↦ suc ∪ 𝑥) ≼ 𝐵))
70	54, 69	mpi 20	. . . . . . . 8 ⊢ (𝐵 ≺ (cf‘𝐴) → ran (𝑥 ∈ 𝐵 ↦ suc ∪ 𝑥) ≼ 𝐵)
71	51, 70	eqbrtrrid 5160	. . . . . . 7 ⊢ (𝐵 ≺ (cf‘𝐴) → {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑦 = suc ∪ 𝑥} ≼ 𝐵)
72		domtr 9026	. . . . . . 7 ⊢ (((cf‘𝐴) ≼ {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑦 = suc ∪ 𝑥} ∧ {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑦 = suc ∪ 𝑥} ≼ 𝐵) → (cf‘𝐴) ≼ 𝐵)
73	71, 72	sylan2 593	. . . . . 6 ⊢ (((cf‘𝐴) ≼ {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑦 = suc ∪ 𝑥} ∧ 𝐵 ≺ (cf‘𝐴)) → (cf‘𝐴) ≼ 𝐵)
74		domnsym 9118	. . . . . 6 ⊢ ((cf‘𝐴) ≼ 𝐵 → ¬ 𝐵 ≺ (cf‘𝐴))
75	73, 74	syl 17	. . . . 5 ⊢ (((cf‘𝐴) ≼ {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑦 = suc ∪ 𝑥} ∧ 𝐵 ≺ (cf‘𝐴)) → ¬ 𝐵 ≺ (cf‘𝐴))
76	75	pm2.01da 798	. . . 4 ⊢ ((cf‘𝐴) ≼ {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑦 = suc ∪ 𝑥} → ¬ 𝐵 ≺ (cf‘𝐴))
77	76	a1i 11	. . 3 ⊢ ((Lim 𝐴 ∧ ∀𝑥 ∈ 𝐵 (𝑥 ⊆ 𝐴 ∧ 𝑥 ≺ (cf‘𝐴))) → ((cf‘𝐴) ≼ {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑦 = suc ∪ 𝑥} → ¬ 𝐵 ≺ (cf‘𝐴)))
78	29, 49, 77	3syld 60	. 2 ⊢ ((Lim 𝐴 ∧ ∀𝑥 ∈ 𝐵 (𝑥 ⊆ 𝐴 ∧ 𝑥 ≺ (cf‘𝐴))) → (∪ 𝐵 = 𝐴 → ¬ 𝐵 ≺ (cf‘𝐴)))
79	78	necon2ad 2948	1 ⊢ ((Lim 𝐴 ∧ ∀𝑥 ∈ 𝐵 (𝑥 ⊆ 𝐴 ∧ 𝑥 ≺ (cf‘𝐴))) → (𝐵 ≺ (cf‘𝐴) → ∪ 𝐵 ≠ 𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  {cab 2714   ≠ wne 2933  ∀wral 3052  ∃wrex 3061  Vcvv 3464   ⊆ wss 3931  ∪ cuni 4888  ∪ ciun 4972   class class class wbr 5124   ↦ cmpt 5206  dom cdm 5659  ran crn 5660  Ord word 6356  Oncon0 6357  Lim wlim 6358  suc csuc 6359   Fn wfn 6531  –onto→wfo 6534  ‘cfv 6536   ≼ cdom 8962   ≺ csdm 8963  cardccrd 9954  cfccf 9956

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 2708  ax-rep 5254  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734

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 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rmo 3364  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-int 4928  df-iun 4974  df-iin 4975  df-br 5125  df-opab 5187  df-mpt 5207  df-tr 5235  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-se 5612  df-we 5613  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6295  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-isom 6545  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-1st 7993  df-2nd 7994  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-er 8724  df-map 8847  df-en 8965  df-dom 8966  df-sdom 8967  df-card 9958  df-cf 9960  df-acn 9961

This theorem is referenced by:  tskuni  10802