Theorem cfslb2n 10248

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 10246. 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 6419	. . . . . . . . . 10 ⊢ (Lim 𝐴 → Ord 𝐴)
2		ordsson 7778	. . . . . . . . . 10 ⊢ (Ord 𝐴 → 𝐴 ⊆ On)
3		sstr 3953	. . . . . . . . . . 11 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝐴 ⊆ On) → 𝑥 ⊆ On)
4	3	expcom 418	. . . . . . . . . 10 ⊢ (𝐴 ⊆ On → (𝑥 ⊆ 𝐴 → 𝑥 ⊆ On))
5	1, 2, 4	3syl 19	. . . . . . . . 9 ⊢ (Lim 𝐴 → (𝑥 ⊆ 𝐴 → 𝑥 ⊆ On))
6		onsucuni 7820	. . . . . . . . 9 ⊢ (𝑥 ⊆ On → 𝑥 ⊆ suc ∪ 𝑥)
7	5, 6	syl6 36	. . . . . . . 8 ⊢ (Lim 𝐴 → (𝑥 ⊆ 𝐴 → 𝑥 ⊆ suc ∪ 𝑥))
8	7	adantrd 496	. . . . . . 7 ⊢ (Lim 𝐴 → ((𝑥 ⊆ 𝐴 ∧ 𝑥 ≺ (cf‘𝐴)) → 𝑥 ⊆ suc ∪ 𝑥))
9	8	ralimdv 3185	. . . . . 6 ⊢ (Lim 𝐴 → (∀𝑥 ∈ 𝐵 (𝑥 ⊆ 𝐴 ∧ 𝑥 ≺ (cf‘𝐴)) → ∀𝑥 ∈ 𝐵 𝑥 ⊆ suc ∪ 𝑥))
10		uniiun 5024	. . . . . . 7 ⊢ ∪ 𝐵 = ∪ 𝑥 ∈ 𝐵 𝑥
11		ss2iun 4976	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐵 𝑥 ⊆ suc ∪ 𝑥 → ∪ 𝑥 ∈ 𝐵 𝑥 ⊆ ∪ 𝑥 ∈ 𝐵 suc ∪ 𝑥)
12	10, 11	eqsstrid 3983	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐵 𝑥 ⊆ suc ∪ 𝑥 → ∪ 𝐵 ⊆ ∪ 𝑥 ∈ 𝐵 suc ∪ 𝑥)
13	9, 12	syl6 36	. . . . 5 ⊢ (Lim 𝐴 → (∀𝑥 ∈ 𝐵 (𝑥 ⊆ 𝐴 ∧ 𝑥 ≺ (cf‘𝐴)) → ∪ 𝐵 ⊆ ∪ 𝑥 ∈ 𝐵 suc ∪ 𝑥))
14	13	imp 411	. . . 4 ⊢ ((Lim 𝐴 ∧ ∀𝑥 ∈ 𝐵 (𝑥 ⊆ 𝐴 ∧ 𝑥 ≺ (cf‘𝐴))) → ∪ 𝐵 ⊆ ∪ 𝑥 ∈ 𝐵 suc ∪ 𝑥)
15		cfslb.1	. . . . . . . . . 10 ⊢ 𝐴 ∈ V
16	15	cfslbn 10247	. . . . . . . . 9 ⊢ ((Lim 𝐴 ∧ 𝑥 ⊆ 𝐴 ∧ 𝑥 ≺ (cf‘𝐴)) → ∪ 𝑥 ∈ 𝐴)
17	16	3expib 1138	. . . . . . . 8 ⊢ (Lim 𝐴 → ((𝑥 ⊆ 𝐴 ∧ 𝑥 ≺ (cf‘𝐴)) → ∪ 𝑥 ∈ 𝐴))
18		ordsucss 7810	. . . . . . . 8 ⊢ (Ord 𝐴 → (∪ 𝑥 ∈ 𝐴 → suc ∪ 𝑥 ⊆ 𝐴))
19	1, 17, 18	sylsyld 62	. . . . . . 7 ⊢ (Lim 𝐴 → ((𝑥 ⊆ 𝐴 ∧ 𝑥 ≺ (cf‘𝐴)) → suc ∪ 𝑥 ⊆ 𝐴))
20	19	ralimdv 3185	. . . . . 6 ⊢ (Lim 𝐴 → (∀𝑥 ∈ 𝐵 (𝑥 ⊆ 𝐴 ∧ 𝑥 ≺ (cf‘𝐴)) → ∀𝑥 ∈ 𝐵 suc ∪ 𝑥 ⊆ 𝐴))
21		iunss 5010	. . . . . 6 ⊢ (∪ 𝑥 ∈ 𝐵 suc ∪ 𝑥 ⊆ 𝐴 ↔ ∀𝑥 ∈ 𝐵 suc ∪ 𝑥 ⊆ 𝐴)
22	20, 21	imbitrrdi 255	. . . . 5 ⊢ (Lim 𝐴 → (∀𝑥 ∈ 𝐵 (𝑥 ⊆ 𝐴 ∧ 𝑥 ≺ (cf‘𝐴)) → ∪ 𝑥 ∈ 𝐵 suc ∪ 𝑥 ⊆ 𝐴))
23	22	imp 411	. . . 4 ⊢ ((Lim 𝐴 ∧ ∀𝑥 ∈ 𝐵 (𝑥 ⊆ 𝐴 ∧ 𝑥 ≺ (cf‘𝐴))) → ∪ 𝑥 ∈ 𝐵 suc ∪ 𝑥 ⊆ 𝐴)
24		sseq1 3970	. . . . . 6 ⊢ (∪ 𝐵 = 𝐴 → (∪ 𝐵 ⊆ ∪ 𝑥 ∈ 𝐵 suc ∪ 𝑥 ↔ 𝐴 ⊆ ∪ 𝑥 ∈ 𝐵 suc ∪ 𝑥))
25		eqss 3960	. . . . . . 7 ⊢ (∪ 𝑥 ∈ 𝐵 suc ∪ 𝑥 = 𝐴 ↔ (∪ 𝑥 ∈ 𝐵 suc ∪ 𝑥 ⊆ 𝐴 ∧ 𝐴 ⊆ ∪ 𝑥 ∈ 𝐵 suc ∪ 𝑥))
26	25	simplbi2com 507	. . . . . 6 ⊢ (𝐴 ⊆ ∪ 𝑥 ∈ 𝐵 suc ∪ 𝑥 → (∪ 𝑥 ∈ 𝐵 suc ∪ 𝑥 ⊆ 𝐴 → ∪ 𝑥 ∈ 𝐵 suc ∪ 𝑥 = 𝐴))
27	24, 26	biimtrdi 256	. . . . 5 ⊢ (∪ 𝐵 = 𝐴 → (∪ 𝐵 ⊆ ∪ 𝑥 ∈ 𝐵 suc ∪ 𝑥 → (∪ 𝑥 ∈ 𝐵 suc ∪ 𝑥 ⊆ 𝐴 → ∪ 𝑥 ∈ 𝐵 suc ∪ 𝑥 = 𝐴)))
28	27	com3l 90	. . . 4 ⊢ (∪ 𝐵 ⊆ ∪ 𝑥 ∈ 𝐵 suc ∪ 𝑥 → (∪ 𝑥 ∈ 𝐵 suc ∪ 𝑥 ⊆ 𝐴 → (∪ 𝐵 = 𝐴 → ∪ 𝑥 ∈ 𝐵 suc ∪ 𝑥 = 𝐴)))
29	14, 23, 28	sylc 66	. . 3 ⊢ ((Lim 𝐴 ∧ ∀𝑥 ∈ 𝐵 (𝑥 ⊆ 𝐴 ∧ 𝑥 ≺ (cf‘𝐴))) → (∪ 𝐵 = 𝐴 → ∪ 𝑥 ∈ 𝐵 suc ∪ 𝑥 = 𝐴))
30		limsuc 7841	. . . . . . . . 9 ⊢ (Lim 𝐴 → (∪ 𝑥 ∈ 𝐴 ↔ suc ∪ 𝑥 ∈ 𝐴))
31	17, 30	sylibd 242	. . . . . . . 8 ⊢ (Lim 𝐴 → ((𝑥 ⊆ 𝐴 ∧ 𝑥 ≺ (cf‘𝐴)) → suc ∪ 𝑥 ∈ 𝐴))
32	31	ralimdv 3185	. . . . . . 7 ⊢ (Lim 𝐴 → (∀𝑥 ∈ 𝐵 (𝑥 ⊆ 𝐴 ∧ 𝑥 ≺ (cf‘𝐴)) → ∀𝑥 ∈ 𝐵 suc ∪ 𝑥 ∈ 𝐴))
33	32	imp 411	. . . . . 6 ⊢ ((Lim 𝐴 ∧ ∀𝑥 ∈ 𝐵 (𝑥 ⊆ 𝐴 ∧ 𝑥 ≺ (cf‘𝐴))) → ∀𝑥 ∈ 𝐵 suc ∪ 𝑥 ∈ 𝐴)
34		r19.29 3134	. . . . . . . 8 ⊢ ((∀𝑥 ∈ 𝐵 suc ∪ 𝑥 ∈ 𝐴 ∧ ∃𝑥 ∈ 𝐵 𝑦 = suc ∪ 𝑥) → ∃𝑥 ∈ 𝐵 (suc ∪ 𝑥 ∈ 𝐴 ∧ 𝑦 = suc ∪ 𝑥))
35		eleq1 2857	. . . . . . . . . 10 ⊢ (𝑦 = suc ∪ 𝑥 → (𝑦 ∈ 𝐴 ↔ suc ∪ 𝑥 ∈ 𝐴))
36	35	biimparc 484	. . . . . . . . 9 ⊢ ((suc ∪ 𝑥 ∈ 𝐴 ∧ 𝑦 = suc ∪ 𝑥) → 𝑦 ∈ 𝐴)
37	36	rexlimivw 3168	. . . . . . . 8 ⊢ (∃𝑥 ∈ 𝐵 (suc ∪ 𝑥 ∈ 𝐴 ∧ 𝑦 = suc ∪ 𝑥) → 𝑦 ∈ 𝐴)
38	34, 37	syl 18	. . . . . . 7 ⊢ ((∀𝑥 ∈ 𝐵 suc ∪ 𝑥 ∈ 𝐴 ∧ ∃𝑥 ∈ 𝐵 𝑦 = suc ∪ 𝑥) → 𝑦 ∈ 𝐴)
39	38	ex 417	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐵 suc ∪ 𝑥 ∈ 𝐴 → (∃𝑥 ∈ 𝐵 𝑦 = suc ∪ 𝑥 → 𝑦 ∈ 𝐴))
40	33, 39	syl 18	. . . . 5 ⊢ ((Lim 𝐴 ∧ ∀𝑥 ∈ 𝐵 (𝑥 ⊆ 𝐴 ∧ 𝑥 ≺ (cf‘𝐴))) → (∃𝑥 ∈ 𝐵 𝑦 = suc ∪ 𝑥 → 𝑦 ∈ 𝐴))
41	40	abssdv 4029	. . . 4 ⊢ ((Lim 𝐴 ∧ ∀𝑥 ∈ 𝐵 (𝑥 ⊆ 𝐴 ∧ 𝑥 ≺ (cf‘𝐴))) → {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑦 = suc ∪ 𝑥} ⊆ 𝐴)
42		vuniex 7734	. . . . . . . 8 ⊢ ∪ 𝑥 ∈ V
43	42	sucex 7801	. . . . . . 7 ⊢ suc ∪ 𝑥 ∈ V
44	43	dfiun2 4997	. . . . . 6 ⊢ ∪ 𝑥 ∈ 𝐵 suc ∪ 𝑥 = ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑦 = suc ∪ 𝑥}
45	44	eqeq1i 2774	. . . . 5 ⊢ (∪ 𝑥 ∈ 𝐵 suc ∪ 𝑥 = 𝐴 ↔ ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑦 = suc ∪ 𝑥} = 𝐴)
46	15	cfslb 10246	. . . . . 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 1137	. . . . 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 245	. . . 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 602	. . 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 2769	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐵 ↦ suc ∪ 𝑥) = (𝑥 ∈ 𝐵 ↦ suc ∪ 𝑥)
51	50	rnmpt 5945	. . . . . . . 8 ⊢ ran (𝑥 ∈ 𝐵 ↦ suc ∪ 𝑥) = {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑦 = suc ∪ 𝑥}
52	43, 50	fnmpti 6676	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐵 ↦ suc ∪ 𝑥) Fn 𝐵
53		dffn4 6796	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐵 ↦ suc ∪ 𝑥) Fn 𝐵 ↔ (𝑥 ∈ 𝐵 ↦ suc ∪ 𝑥):𝐵–onto→ran (𝑥 ∈ 𝐵 ↦ suc ∪ 𝑥))
54	52, 53	mpbi 233	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐵 ↦ suc ∪ 𝑥):𝐵–onto→ran (𝑥 ∈ 𝐵 ↦ suc ∪ 𝑥)
55		relsdom 8946	. . . . . . . . . . 11 ⊢ Rel ≺
56	55	brrelex1i 5715	. . . . . . . . . 10 ⊢ (𝐵 ≺ (cf‘𝐴) → 𝐵 ∈ V)
57		breq1 5113	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝐵 → (𝑦 ≺ (cf‘𝐴) ↔ 𝐵 ≺ (cf‘𝐴)))
58		foeq2 6787	. . . . . . . . . . . . 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 5114	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝐵 → (ran (𝑥 ∈ 𝐵 ↦ suc ∪ 𝑥) ≼ 𝑦 ↔ ran (𝑥 ∈ 𝐵 ↦ suc ∪ 𝑥) ≼ 𝐵))
60	58, 59	imbi12d 347	. . . . . . . . . . . 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 347	. . . . . . . . . . 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 10234	. . . . . . . . . . . . 13 ⊢ (cf‘𝐴) ∈ On
63		sdomdom 8973	. . . . . . . . . . . . 13 ⊢ (𝑦 ≺ (cf‘𝐴) → 𝑦 ≼ (cf‘𝐴))
64		ondomen 10017	. . . . . . . . . . . . 13 ⊢ (((cf‘𝐴) ∈ On ∧ 𝑦 ≼ (cf‘𝐴)) → 𝑦 ∈ dom card)
65	62, 63, 64	sylancr 598	. . . . . . . . . . . 12 ⊢ (𝑦 ≺ (cf‘𝐴) → 𝑦 ∈ dom card)
66		fodomnum 10037	. . . . . . . . . . . 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 18	. . . . . . . . . . 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 3531	. . . . . . . . . 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 39	. . . . . . . . 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 21	. . . . . . . 8 ⊢ (𝐵 ≺ (cf‘𝐴) → ran (𝑥 ∈ 𝐵 ↦ suc ∪ 𝑥) ≼ 𝐵)
71	51, 70	eqbrtrrid 5148	. . . . . . 7 ⊢ (𝐵 ≺ (cf‘𝐴) → {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑦 = suc ∪ 𝑥} ≼ 𝐵)
72		domtr 9000	. . . . . . 7 ⊢ (((cf‘𝐴) ≼ {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑦 = suc ∪ 𝑥} ∧ {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑦 = suc ∪ 𝑥} ≼ 𝐵) → (cf‘𝐴) ≼ 𝐵)
73	71, 72	sylan2 604	. . . . . 6 ⊢ (((cf‘𝐴) ≼ {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑦 = suc ∪ 𝑥} ∧ 𝐵 ≺ (cf‘𝐴)) → (cf‘𝐴) ≼ 𝐵)
74		domnsym 9087	. . . . . 6 ⊢ ((cf‘𝐴) ≼ 𝐵 → ¬ 𝐵 ≺ (cf‘𝐴))
75	73, 74	syl 18	. . . . 5 ⊢ (((cf‘𝐴) ≼ {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑦 = suc ∪ 𝑥} ∧ 𝐵 ≺ (cf‘𝐴)) → ¬ 𝐵 ≺ (cf‘𝐴))
76	75	pm2.01da 810	. . . 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 61	. 2 ⊢ ((Lim 𝐴 ∧ ∀𝑥 ∈ 𝐵 (𝑥 ⊆ 𝐴 ∧ 𝑥 ≺ (cf‘𝐴))) → (∪ 𝐵 = 𝐴 → ¬ 𝐵 ≺ (cf‘𝐴)))
79	78	necon2ad 2979	1 ⊢ ((Lim 𝐴 ∧ ∀𝑥 ∈ 𝐵 (𝑥 ⊆ 𝐴 ∧ 𝑥 ≺ (cf‘𝐴))) → (𝐵 ≺ (cf‘𝐴) → ∪ 𝐵 ≠ 𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 400   = wceq 1567   ∈ wcel 2149  {cab 2747   ≠ wne 2964  ∀wral 3085  ∃wrex 3095  Vcvv 3463   ⊆ wss 3913  ∪ cuni 4873  ∪ ciun 4957   class class class wbr 5110   ↦ cmpt 5193  dom cdm 5659  ran crn 5660  Ord word 6356  Oncon0 6357  Lim wlim 6358  suc csuc 6359   Fn wfn 6528  –onto→wfo 6531  ‘cfv 6533   ≼ cdom 8937   ≺ csdm 8938  cardccrd 9917  cfccf 9919

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 5239  ax-sep 5258  ax-nul 5268  ax-pow 5334  ax-pr 5402  ax-un 7730

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 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-int 4914  df-iun 4959  df-iin 4960  df-br 5111  df-opab 5175  df-mpt 5194  df-tr 5220  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-se 5613  df-we 5614  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 6299  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-isom 6542  df-riota 7365  df-ov 7411  df-oprab 7412  df-mpo 7413  df-1st 7982  df-2nd 7983  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-er 8690  df-map 8822  df-en 8940  df-dom 8941  df-sdom 8942  df-card 9921  df-cf 9923  df-acn 9924

This theorem is referenced by:  tskuni  10764