Theorem cfslb2n 10219

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 10217. 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 6402	. . . . . . . . . 10 ⊢ (Lim 𝐴 → Ord 𝐴)
2		ordsson 7761	. . . . . . . . . 10 ⊢ (Ord 𝐴 → 𝐴 ⊆ On)
3		sstr 3942	. . . . . . . . . . 11 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝐴 ⊆ On) → 𝑥 ⊆ On)
4	3	expcom 417	. . . . . . . . . 10 ⊢ (𝐴 ⊆ On → (𝑥 ⊆ 𝐴 → 𝑥 ⊆ On))
5	1, 2, 4	3syl 18	. . . . . . . . 9 ⊢ (Lim 𝐴 → (𝑥 ⊆ 𝐴 → 𝑥 ⊆ On))
6		onsucuni 7803	. . . . . . . . 9 ⊢ (𝑥 ⊆ On → 𝑥 ⊆ suc ∪ 𝑥)
7	5, 6	syl6 35	. . . . . . . 8 ⊢ (Lim 𝐴 → (𝑥 ⊆ 𝐴 → 𝑥 ⊆ suc ∪ 𝑥))
8	7	adantrd 495	. . . . . . 7 ⊢ (Lim 𝐴 → ((𝑥 ⊆ 𝐴 ∧ 𝑥 ≺ (cf‘𝐴)) → 𝑥 ⊆ suc ∪ 𝑥))
9	8	ralimdv 3175	. . . . . 6 ⊢ (Lim 𝐴 → (∀𝑥 ∈ 𝐵 (𝑥 ⊆ 𝐴 ∧ 𝑥 ≺ (cf‘𝐴)) → ∀𝑥 ∈ 𝐵 𝑥 ⊆ suc ∪ 𝑥))
10		uniiun 5013	. . . . . . 7 ⊢ ∪ 𝐵 = ∪ 𝑥 ∈ 𝐵 𝑥
11		ss2iun 4965	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐵 𝑥 ⊆ suc ∪ 𝑥 → ∪ 𝑥 ∈ 𝐵 𝑥 ⊆ ∪ 𝑥 ∈ 𝐵 suc ∪ 𝑥)
12	10, 11	eqsstrid 3972	. . . . . 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 410	. . . 4 ⊢ ((Lim 𝐴 ∧ ∀𝑥 ∈ 𝐵 (𝑥 ⊆ 𝐴 ∧ 𝑥 ≺ (cf‘𝐴))) → ∪ 𝐵 ⊆ ∪ 𝑥 ∈ 𝐵 suc ∪ 𝑥)
15		cfslb.1	. . . . . . . . . 10 ⊢ 𝐴 ∈ V
16	15	cfslbn 10218	. . . . . . . . 9 ⊢ ((Lim 𝐴 ∧ 𝑥 ⊆ 𝐴 ∧ 𝑥 ≺ (cf‘𝐴)) → ∪ 𝑥 ∈ 𝐴)
17	16	3expib 1134	. . . . . . . 8 ⊢ (Lim 𝐴 → ((𝑥 ⊆ 𝐴 ∧ 𝑥 ≺ (cf‘𝐴)) → ∪ 𝑥 ∈ 𝐴))
18		ordsucss 7793	. . . . . . . 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 3175	. . . . . 6 ⊢ (Lim 𝐴 → (∀𝑥 ∈ 𝐵 (𝑥 ⊆ 𝐴 ∧ 𝑥 ≺ (cf‘𝐴)) → ∀𝑥 ∈ 𝐵 suc ∪ 𝑥 ⊆ 𝐴))
21		iunss 4999	. . . . . 6 ⊢ (∪ 𝑥 ∈ 𝐵 suc ∪ 𝑥 ⊆ 𝐴 ↔ ∀𝑥 ∈ 𝐵 suc ∪ 𝑥 ⊆ 𝐴)
22	20, 21	imbitrrdi 254	. . . . 5 ⊢ (Lim 𝐴 → (∀𝑥 ∈ 𝐵 (𝑥 ⊆ 𝐴 ∧ 𝑥 ≺ (cf‘𝐴)) → ∪ 𝑥 ∈ 𝐵 suc ∪ 𝑥 ⊆ 𝐴))
23	22	imp 410	. . . 4 ⊢ ((Lim 𝐴 ∧ ∀𝑥 ∈ 𝐵 (𝑥 ⊆ 𝐴 ∧ 𝑥 ≺ (cf‘𝐴))) → ∪ 𝑥 ∈ 𝐵 suc ∪ 𝑥 ⊆ 𝐴)
24		sseq1 3959	. . . . . 6 ⊢ (∪ 𝐵 = 𝐴 → (∪ 𝐵 ⊆ ∪ 𝑥 ∈ 𝐵 suc ∪ 𝑥 ↔ 𝐴 ⊆ ∪ 𝑥 ∈ 𝐵 suc ∪ 𝑥))
25		eqss 3949	. . . . . . 7 ⊢ (∪ 𝑥 ∈ 𝐵 suc ∪ 𝑥 = 𝐴 ↔ (∪ 𝑥 ∈ 𝐵 suc ∪ 𝑥 ⊆ 𝐴 ∧ 𝐴 ⊆ ∪ 𝑥 ∈ 𝐵 suc ∪ 𝑥))
26	25	simplbi2com 506	. . . . . 6 ⊢ (𝐴 ⊆ ∪ 𝑥 ∈ 𝐵 suc ∪ 𝑥 → (∪ 𝑥 ∈ 𝐵 suc ∪ 𝑥 ⊆ 𝐴 → ∪ 𝑥 ∈ 𝐵 suc ∪ 𝑥 = 𝐴))
27	24, 26	biimtrdi 255	. . . . 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 7824	. . . . . . . . 9 ⊢ (Lim 𝐴 → (∪ 𝑥 ∈ 𝐴 ↔ suc ∪ 𝑥 ∈ 𝐴))
31	17, 30	sylibd 241	. . . . . . . 8 ⊢ (Lim 𝐴 → ((𝑥 ⊆ 𝐴 ∧ 𝑥 ≺ (cf‘𝐴)) → suc ∪ 𝑥 ∈ 𝐴))
32	31	ralimdv 3175	. . . . . . 7 ⊢ (Lim 𝐴 → (∀𝑥 ∈ 𝐵 (𝑥 ⊆ 𝐴 ∧ 𝑥 ≺ (cf‘𝐴)) → ∀𝑥 ∈ 𝐵 suc ∪ 𝑥 ∈ 𝐴))
33	32	imp 410	. . . . . 6 ⊢ ((Lim 𝐴 ∧ ∀𝑥 ∈ 𝐵 (𝑥 ⊆ 𝐴 ∧ 𝑥 ≺ (cf‘𝐴))) → ∀𝑥 ∈ 𝐵 suc ∪ 𝑥 ∈ 𝐴)
34		r19.29 3124	. . . . . . . 8 ⊢ ((∀𝑥 ∈ 𝐵 suc ∪ 𝑥 ∈ 𝐴 ∧ ∃𝑥 ∈ 𝐵 𝑦 = suc ∪ 𝑥) → ∃𝑥 ∈ 𝐵 (suc ∪ 𝑥 ∈ 𝐴 ∧ 𝑦 = suc ∪ 𝑥))
35		eleq1 2849	. . . . . . . . . 10 ⊢ (𝑦 = suc ∪ 𝑥 → (𝑦 ∈ 𝐴 ↔ suc ∪ 𝑥 ∈ 𝐴))
36	35	biimparc 483	. . . . . . . . 9 ⊢ ((suc ∪ 𝑥 ∈ 𝐴 ∧ 𝑦 = suc ∪ 𝑥) → 𝑦 ∈ 𝐴)
37	36	rexlimivw 3158	. . . . . . . 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 416	. . . . . 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 4018	. . . 4 ⊢ ((Lim 𝐴 ∧ ∀𝑥 ∈ 𝐵 (𝑥 ⊆ 𝐴 ∧ 𝑥 ≺ (cf‘𝐴))) → {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑦 = suc ∪ 𝑥} ⊆ 𝐴)
42		vuniex 7717	. . . . . . . 8 ⊢ ∪ 𝑥 ∈ V
43	42	sucex 7784	. . . . . . 7 ⊢ suc ∪ 𝑥 ∈ V
44	43	dfiun2 4986	. . . . . 6 ⊢ ∪ 𝑥 ∈ 𝐵 suc ∪ 𝑥 = ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑦 = suc ∪ 𝑥}
45	44	eqeq1i 2766	. . . . 5 ⊢ (∪ 𝑥 ∈ 𝐵 suc ∪ 𝑥 = 𝐴 ↔ ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑦 = suc ∪ 𝑥} = 𝐴)
46	15	cfslb 10217	. . . . . 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 1133	. . . . 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 244	. . . 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 600	. . 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 2761	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐵 ↦ suc ∪ 𝑥) = (𝑥 ∈ 𝐵 ↦ suc ∪ 𝑥)
51	50	rnmpt 5929	. . . . . . . 8 ⊢ ran (𝑥 ∈ 𝐵 ↦ suc ∪ 𝑥) = {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑦 = suc ∪ 𝑥}
52	43, 50	fnmpti 6659	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐵 ↦ suc ∪ 𝑥) Fn 𝐵
53		dffn4 6779	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐵 ↦ suc ∪ 𝑥) Fn 𝐵 ↔ (𝑥 ∈ 𝐵 ↦ suc ∪ 𝑥):𝐵–onto→ran (𝑥 ∈ 𝐵 ↦ suc ∪ 𝑥))
54	52, 53	mpbi 232	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐵 ↦ suc ∪ 𝑥):𝐵–onto→ran (𝑥 ∈ 𝐵 ↦ suc ∪ 𝑥)
55		relsdom 8928	. . . . . . . . . . 11 ⊢ Rel ≺
56	55	brrelex1i 5699	. . . . . . . . . 10 ⊢ (𝐵 ≺ (cf‘𝐴) → 𝐵 ∈ V)
57		breq1 5100	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝐵 → (𝑦 ≺ (cf‘𝐴) ↔ 𝐵 ≺ (cf‘𝐴)))
58		foeq2 6770	. . . . . . . . . . . . 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 5101	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝐵 → (ran (𝑥 ∈ 𝐵 ↦ suc ∪ 𝑥) ≼ 𝑦 ↔ ran (𝑥 ∈ 𝐵 ↦ suc ∪ 𝑥) ≼ 𝐵))
60	58, 59	imbi12d 346	. . . . . . . . . . . 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 346	. . . . . . . . . . 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 10205	. . . . . . . . . . . . 13 ⊢ (cf‘𝐴) ∈ On
63		sdomdom 8955	. . . . . . . . . . . . 13 ⊢ (𝑦 ≺ (cf‘𝐴) → 𝑦 ≼ (cf‘𝐴))
64		ondomen 9987	. . . . . . . . . . . . 13 ⊢ (((cf‘𝐴) ∈ On ∧ 𝑦 ≼ (cf‘𝐴)) → 𝑦 ∈ dom card)
65	62, 63, 64	sylancr 596	. . . . . . . . . . . 12 ⊢ (𝑦 ≺ (cf‘𝐴) → 𝑦 ∈ dom card)
66		fodomnum 10007	. . . . . . . . . . . 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 3521	. . . . . . . . . 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 5133	. . . . . . 7 ⊢ (𝐵 ≺ (cf‘𝐴) → {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑦 = suc ∪ 𝑥} ≼ 𝐵)
72		domtr 8982	. . . . . . 7 ⊢ (((cf‘𝐴) ≼ {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑦 = suc ∪ 𝑥} ∧ {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑦 = suc ∪ 𝑥} ≼ 𝐵) → (cf‘𝐴) ≼ 𝐵)
73	71, 72	sylan2 602	. . . . . 6 ⊢ (((cf‘𝐴) ≼ {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑦 = suc ∪ 𝑥} ∧ 𝐵 ≺ (cf‘𝐴)) → (cf‘𝐴) ≼ 𝐵)
74		domnsym 9069	. . . . . 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 808	. . . 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 2971	1 ⊢ ((Lim 𝐴 ∧ ∀𝑥 ∈ 𝐵 (𝑥 ⊆ 𝐴 ∧ 𝑥 ≺ (cf‘𝐴))) → (𝐵 ≺ (cf‘𝐴) → ∪ 𝐵 ≠ 𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   = wceq 1559   ∈ wcel 2141  {cab 2739   ≠ wne 2956  ∀wral 3075  ∃wrex 3085  Vcvv 3453   ⊆ wss 3902  ∪ cuni 4862  ∪ ciun 4946   class class class wbr 5097   ↦ cmpt 5178  dom cdm 5643  ran crn 5644  Ord word 6340  Oncon0 6341  Lim wlim 6342  suc csuc 6343   Fn wfn 6511  –onto→wfo 6514  ‘cfv 6516   ≼ cdom 8919   ≺ csdm 8920  cardccrd 9887  cfccf 9889

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5224  ax-sep 5243  ax-nul 5253  ax-pow 5319  ax-pr 5387  ax-un 7713

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-int 4903  df-iun 4948  df-iin 4949  df-br 5098  df-opab 5160  df-mpt 5179  df-tr 5205  df-id 5538  df-eprel 5543  df-po 5551  df-so 5552  df-fr 5596  df-se 5597  df-we 5598  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-res 5655  df-ima 5656  df-pred 6283  df-ord 6344  df-on 6345  df-lim 6346  df-suc 6347  df-iota 6472  df-fun 6518  df-fn 6519  df-f 6520  df-f1 6521  df-fo 6522  df-f1o 6523  df-fv 6524  df-isom 6525  df-riota 7348  df-ov 7394  df-oprab 7395  df-mpo 7396  df-1st 7965  df-2nd 7966  df-frecs 8256  df-wrecs 8287  df-recs 8336  df-er 8672  df-map 8804  df-en 8922  df-dom 8923  df-sdom 8924  df-card 9891  df-cf 9893  df-acn 9894

This theorem is referenced by:  tskuni  10735