Theorem isfin2-2 10304

Description: Fin^II expressed in terms of minimal elements. (Contributed by Stefan O'Rear, 2-Nov-2014.) (Proof shortened by Mario Carneiro, 16-May-2015.)

Assertion

Ref	Expression
isfin2-2	⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ FinII ↔ ∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [⊊] Or 𝑦) → ∩ 𝑦 ∈ 𝑦)))

Distinct variable group:   𝑦,𝐴

Allowed substitution hint:   𝑉(𝑦)

Proof of Theorem isfin2-2

Dummy variables 𝑏 𝑐 𝑚 𝑛 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elpwi 4570	. . . 4 ⊢ (𝑦 ∈ 𝒫 𝒫 𝐴 → 𝑦 ⊆ 𝒫 𝐴)
2		fin2i2 10303	. . . . 5 ⊢ (((𝐴 ∈ FinII ∧ 𝑦 ⊆ 𝒫 𝐴) ∧ (𝑦 ≠ ∅ ∧ [⊊] Or 𝑦)) → ∩ 𝑦 ∈ 𝑦)
3	2	ex 417	. . . 4 ⊢ ((𝐴 ∈ FinII ∧ 𝑦 ⊆ 𝒫 𝐴) → ((𝑦 ≠ ∅ ∧ [⊊] Or 𝑦) → ∩ 𝑦 ∈ 𝑦))
4	1, 3	sylan2 604	. . 3 ⊢ ((𝐴 ∈ FinII ∧ 𝑦 ∈ 𝒫 𝒫 𝐴) → ((𝑦 ≠ ∅ ∧ [⊊] Or 𝑦) → ∩ 𝑦 ∈ 𝑦))
5	4	ralrimiva 3157	. 2 ⊢ (𝐴 ∈ FinII → ∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [⊊] Or 𝑦) → ∩ 𝑦 ∈ 𝑦))
6		elpwi 4570	. . . . 5 ⊢ (𝑏 ∈ 𝒫 𝒫 𝐴 → 𝑏 ⊆ 𝒫 𝐴)
7		simp1r 1217	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑏 ⊆ 𝒫 𝐴) ∧ ∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [⊊] Or 𝑦) → ∩ 𝑦 ∈ 𝑦) ∧ (𝑏 ≠ ∅ ∧ [⊊] Or 𝑏)) → 𝑏 ⊆ 𝒫 𝐴)
8		simp1l 1216	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑏 ⊆ 𝒫 𝐴) ∧ ∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [⊊] Or 𝑦) → ∩ 𝑦 ∈ 𝑦) ∧ (𝑏 ≠ ∅ ∧ [⊊] Or 𝑏)) → 𝐴 ∈ 𝑉)
9		simp3l 1220	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑏 ⊆ 𝒫 𝐴) ∧ ∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [⊊] Or 𝑦) → ∩ 𝑦 ∈ 𝑦) ∧ (𝑏 ≠ ∅ ∧ [⊊] Or 𝑏)) → 𝑏 ≠ ∅)
10		fin23lem7 10301	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑏 ⊆ 𝒫 𝐴 ∧ 𝑏 ≠ ∅) → {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑐) ∈ 𝑏} ≠ ∅)
11	8, 7, 9, 10	syl3anc 1398	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑏 ⊆ 𝒫 𝐴) ∧ ∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [⊊] Or 𝑦) → ∩ 𝑦 ∈ 𝑦) ∧ (𝑏 ≠ ∅ ∧ [⊊] Or 𝑏)) → {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑐) ∈ 𝑏} ≠ ∅)
12		sorpsscmpl 7733	. . . . . . . . . . . 12 ⊢ ( [⊊] Or 𝑏 → [⊊] Or {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑐) ∈ 𝑏})
13	12	adantl 486	. . . . . . . . . . 11 ⊢ ((𝑏 ≠ ∅ ∧ [⊊] Or 𝑏) → [⊊] Or {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑐) ∈ 𝑏})
14	13	3ad2ant3 1153	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑏 ⊆ 𝒫 𝐴) ∧ ∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [⊊] Or 𝑦) → ∩ 𝑦 ∈ 𝑦) ∧ (𝑏 ≠ ∅ ∧ [⊊] Or 𝑏)) → [⊊] Or {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑐) ∈ 𝑏})
15		neeq1 3020	. . . . . . . . . . . . 13 ⊢ (𝑦 = {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑐) ∈ 𝑏} → (𝑦 ≠ ∅ ↔ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑐) ∈ 𝑏} ≠ ∅))
16		soeq2 5593	. . . . . . . . . . . . 13 ⊢ (𝑦 = {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑐) ∈ 𝑏} → ( [⊊] Or 𝑦 ↔ [⊊] Or {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑐) ∈ 𝑏}))
17	15, 16	anbi12d 643	. . . . . . . . . . . 12 ⊢ (𝑦 = {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑐) ∈ 𝑏} → ((𝑦 ≠ ∅ ∧ [⊊] Or 𝑦) ↔ ({𝑐 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑐) ∈ 𝑏} ≠ ∅ ∧ [⊊] Or {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑐) ∈ 𝑏})))
18		inteq 4916	. . . . . . . . . . . . 13 ⊢ (𝑦 = {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑐) ∈ 𝑏} → ∩ 𝑦 = ∩ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑐) ∈ 𝑏})
19		id 23	. . . . . . . . . . . . 13 ⊢ (𝑦 = {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑐) ∈ 𝑏} → 𝑦 = {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑐) ∈ 𝑏})
20	18, 19	eleq12d 2857	. . . . . . . . . . . 12 ⊢ (𝑦 = {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑐) ∈ 𝑏} → (∩ 𝑦 ∈ 𝑦 ↔ ∩ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑐) ∈ 𝑏} ∈ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑐) ∈ 𝑏}))
21	17, 20	imbi12d 347	. . . . . . . . . . 11 ⊢ (𝑦 = {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑐) ∈ 𝑏} → (((𝑦 ≠ ∅ ∧ [⊊] Or 𝑦) → ∩ 𝑦 ∈ 𝑦) ↔ (({𝑐 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑐) ∈ 𝑏} ≠ ∅ ∧ [⊊] Or {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑐) ∈ 𝑏}) → ∩ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑐) ∈ 𝑏} ∈ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑐) ∈ 𝑏})))
22		simp2 1155	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑏 ⊆ 𝒫 𝐴) ∧ ∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [⊊] Or 𝑦) → ∩ 𝑦 ∈ 𝑦) ∧ (𝑏 ≠ ∅ ∧ [⊊] Or 𝑏)) → ∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [⊊] Or 𝑦) → ∩ 𝑦 ∈ 𝑦))
23		ssrab2 4035	. . . . . . . . . . . 12 ⊢ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑐) ∈ 𝑏} ⊆ 𝒫 𝐴
24		pwexg 5351	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ V)
25		elpw2g 5305	. . . . . . . . . . . . 13 ⊢ (𝒫 𝐴 ∈ V → ({𝑐 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑐) ∈ 𝑏} ∈ 𝒫 𝒫 𝐴 ↔ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑐) ∈ 𝑏} ⊆ 𝒫 𝐴))
26	8, 24, 25	3syl 19	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑏 ⊆ 𝒫 𝐴) ∧ ∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [⊊] Or 𝑦) → ∩ 𝑦 ∈ 𝑦) ∧ (𝑏 ≠ ∅ ∧ [⊊] Or 𝑏)) → ({𝑐 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑐) ∈ 𝑏} ∈ 𝒫 𝒫 𝐴 ↔ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑐) ∈ 𝑏} ⊆ 𝒫 𝐴))
27	23, 26	mpbiri 261	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑏 ⊆ 𝒫 𝐴) ∧ ∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [⊊] Or 𝑦) → ∩ 𝑦 ∈ 𝑦) ∧ (𝑏 ≠ ∅ ∧ [⊊] Or 𝑏)) → {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑐) ∈ 𝑏} ∈ 𝒫 𝒫 𝐴)
28	21, 22, 27	rspcdva 3583	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑏 ⊆ 𝒫 𝐴) ∧ ∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [⊊] Or 𝑦) → ∩ 𝑦 ∈ 𝑦) ∧ (𝑏 ≠ ∅ ∧ [⊊] Or 𝑏)) → (({𝑐 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑐) ∈ 𝑏} ≠ ∅ ∧ [⊊] Or {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑐) ∈ 𝑏}) → ∩ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑐) ∈ 𝑏} ∈ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑐) ∈ 𝑏}))
29	11, 14, 28	mp2and 711	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑏 ⊆ 𝒫 𝐴) ∧ ∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [⊊] Or 𝑦) → ∩ 𝑦 ∈ 𝑦) ∧ (𝑏 ≠ ∅ ∧ [⊊] Or 𝑏)) → ∩ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑐) ∈ 𝑏} ∈ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑐) ∈ 𝑏})
30		sorpssint 7732	. . . . . . . . . 10 ⊢ ( [⊊] Or {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑐) ∈ 𝑏} → (∃𝑧 ∈ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑐) ∈ 𝑏}∀𝑤 ∈ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑐) ∈ 𝑏} ¬ 𝑤 ⊊ 𝑧 ↔ ∩ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑐) ∈ 𝑏} ∈ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑐) ∈ 𝑏}))
31	14, 30	syl 18	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑏 ⊆ 𝒫 𝐴) ∧ ∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [⊊] Or 𝑦) → ∩ 𝑦 ∈ 𝑦) ∧ (𝑏 ≠ ∅ ∧ [⊊] Or 𝑏)) → (∃𝑧 ∈ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑐) ∈ 𝑏}∀𝑤 ∈ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑐) ∈ 𝑏} ¬ 𝑤 ⊊ 𝑧 ↔ ∩ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑐) ∈ 𝑏} ∈ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑐) ∈ 𝑏}))
32	29, 31	mpbird 260	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑏 ⊆ 𝒫 𝐴) ∧ ∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [⊊] Or 𝑦) → ∩ 𝑦 ∈ 𝑦) ∧ (𝑏 ≠ ∅ ∧ [⊊] Or 𝑏)) → ∃𝑧 ∈ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑐) ∈ 𝑏}∀𝑤 ∈ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑐) ∈ 𝑏} ¬ 𝑤 ⊊ 𝑧)
33		psseq1 4045	. . . . . . . . 9 ⊢ (𝑚 = (𝐴 ∖ 𝑧) → (𝑚 ⊊ 𝑛 ↔ (𝐴 ∖ 𝑧) ⊊ 𝑛))
34		psseq1 4045	. . . . . . . . 9 ⊢ (𝑤 = (𝐴 ∖ 𝑛) → (𝑤 ⊊ 𝑧 ↔ (𝐴 ∖ 𝑛) ⊊ 𝑧))
35		pssdifcom1 4451	. . . . . . . . 9 ⊢ ((𝑧 ⊆ 𝐴 ∧ 𝑛 ⊆ 𝐴) → ((𝐴 ∖ 𝑧) ⊊ 𝑛 ↔ (𝐴 ∖ 𝑛) ⊊ 𝑧))
36	33, 34, 35	fin23lem11 10302	. . . . . . . 8 ⊢ (𝑏 ⊆ 𝒫 𝐴 → (∃𝑧 ∈ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑐) ∈ 𝑏}∀𝑤 ∈ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑐) ∈ 𝑏} ¬ 𝑤 ⊊ 𝑧 → ∃𝑚 ∈ 𝑏 ∀𝑛 ∈ 𝑏 ¬ 𝑚 ⊊ 𝑛))
37	7, 32, 36	sylc 66	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑏 ⊆ 𝒫 𝐴) ∧ ∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [⊊] Or 𝑦) → ∩ 𝑦 ∈ 𝑦) ∧ (𝑏 ≠ ∅ ∧ [⊊] Or 𝑏)) → ∃𝑚 ∈ 𝑏 ∀𝑛 ∈ 𝑏 ¬ 𝑚 ⊊ 𝑛)
38		simp3r 1221	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑏 ⊆ 𝒫 𝐴) ∧ ∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [⊊] Or 𝑦) → ∩ 𝑦 ∈ 𝑦) ∧ (𝑏 ≠ ∅ ∧ [⊊] Or 𝑏)) → [⊊] Or 𝑏)
39		sorpssuni 7731	. . . . . . . 8 ⊢ ( [⊊] Or 𝑏 → (∃𝑚 ∈ 𝑏 ∀𝑛 ∈ 𝑏 ¬ 𝑚 ⊊ 𝑛 ↔ ∪ 𝑏 ∈ 𝑏))
40	38, 39	syl 18	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑏 ⊆ 𝒫 𝐴) ∧ ∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [⊊] Or 𝑦) → ∩ 𝑦 ∈ 𝑦) ∧ (𝑏 ≠ ∅ ∧ [⊊] Or 𝑏)) → (∃𝑚 ∈ 𝑏 ∀𝑛 ∈ 𝑏 ¬ 𝑚 ⊊ 𝑛 ↔ ∪ 𝑏 ∈ 𝑏))
41	37, 40	mpbid 235	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑏 ⊆ 𝒫 𝐴) ∧ ∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [⊊] Or 𝑦) → ∩ 𝑦 ∈ 𝑦) ∧ (𝑏 ≠ ∅ ∧ [⊊] Or 𝑏)) → ∪ 𝑏 ∈ 𝑏)
42	41	3exp 1137	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑏 ⊆ 𝒫 𝐴) → (∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [⊊] Or 𝑦) → ∩ 𝑦 ∈ 𝑦) → ((𝑏 ≠ ∅ ∧ [⊊] Or 𝑏) → ∪ 𝑏 ∈ 𝑏)))
43	6, 42	sylan2 604	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑏 ∈ 𝒫 𝒫 𝐴) → (∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [⊊] Or 𝑦) → ∩ 𝑦 ∈ 𝑦) → ((𝑏 ≠ ∅ ∧ [⊊] Or 𝑏) → ∪ 𝑏 ∈ 𝑏)))
44	43	ralrimdva 3165	. . 3 ⊢ (𝐴 ∈ 𝑉 → (∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [⊊] Or 𝑦) → ∩ 𝑦 ∈ 𝑦) → ∀𝑏 ∈ 𝒫 𝒫 𝐴((𝑏 ≠ ∅ ∧ [⊊] Or 𝑏) → ∪ 𝑏 ∈ 𝑏)))
45		isfin2 10279	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ FinII ↔ ∀𝑏 ∈ 𝒫 𝒫 𝐴((𝑏 ≠ ∅ ∧ [⊊] Or 𝑏) → ∪ 𝑏 ∈ 𝑏)))
46	44, 45	sylibrd 262	. 2 ⊢ (𝐴 ∈ 𝑉 → (∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [⊊] Or 𝑦) → ∩ 𝑦 ∈ 𝑦) → 𝐴 ∈ FinII))
47	5, 46	impbid2 229	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ FinII ↔ ∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [⊊] Or 𝑦) → ∩ 𝑦 ∈ 𝑦)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143   ≠ wne 2958  ∀wral 3079  ∃wrex 3089  {crab 3416  Vcvv 3455   ∖ cdif 3903   ⊆ wss 3906   ⊊ wpss 3907  ∅c0 4287  𝒫 cpw 4563  ∪ cuni 4873  ∩ cint 4913   Or wor 5570   [⊊] crpss 7721  Fin^IIcfin2 10264

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5258  ax-pow 5338  ax-pr 5406  ax-un 7734

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-int 4914  df-br 5111  df-opab 5175  df-po 5571  df-so 5572  df-xp 5669  df-rel 5670  df-rpss 7722  df-fin2 10271

This theorem is referenced by: (None)