Theorem isfin2-2 9587

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 4463	. . . 4 ⊢ (𝑦 ∈ 𝒫 𝒫 𝐴 → 𝑦 ⊆ 𝒫 𝐴)
2		fin2i2 9586	. . . . 5 ⊢ (((𝐴 ∈ FinII ∧ 𝑦 ⊆ 𝒫 𝐴) ∧ (𝑦 ≠ ∅ ∧ [⊊] Or 𝑦)) → ∩ 𝑦 ∈ 𝑦)
3	2	ex 413	. . . 4 ⊢ ((𝐴 ∈ FinII ∧ 𝑦 ⊆ 𝒫 𝐴) → ((𝑦 ≠ ∅ ∧ [⊊] Or 𝑦) → ∩ 𝑦 ∈ 𝑦))
4	1, 3	sylan2 592	. . 3 ⊢ ((𝐴 ∈ FinII ∧ 𝑦 ∈ 𝒫 𝒫 𝐴) → ((𝑦 ≠ ∅ ∧ [⊊] Or 𝑦) → ∩ 𝑦 ∈ 𝑦))
5	4	ralrimiva 3149	. 2 ⊢ (𝐴 ∈ FinII → ∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [⊊] Or 𝑦) → ∩ 𝑦 ∈ 𝑦))
6		elpwi 4463	. . . . 5 ⊢ (𝑏 ∈ 𝒫 𝒫 𝐴 → 𝑏 ⊆ 𝒫 𝐴)
7		simp1r 1191	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑏 ⊆ 𝒫 𝐴) ∧ ∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [⊊] Or 𝑦) → ∩ 𝑦 ∈ 𝑦) ∧ (𝑏 ≠ ∅ ∧ [⊊] Or 𝑏)) → 𝑏 ⊆ 𝒫 𝐴)
8		simp1l 1190	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑏 ⊆ 𝒫 𝐴) ∧ ∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [⊊] Or 𝑦) → ∩ 𝑦 ∈ 𝑦) ∧ (𝑏 ≠ ∅ ∧ [⊊] Or 𝑏)) → 𝐴 ∈ 𝑉)
9		simp3l 1194	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑏 ⊆ 𝒫 𝐴) ∧ ∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [⊊] Or 𝑦) → ∩ 𝑦 ∈ 𝑦) ∧ (𝑏 ≠ ∅ ∧ [⊊] Or 𝑏)) → 𝑏 ≠ ∅)
10		fin23lem7 9584	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑏 ⊆ 𝒫 𝐴 ∧ 𝑏 ≠ ∅) → {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑐) ∈ 𝑏} ≠ ∅)
11	8, 7, 9, 10	syl3anc 1364	. . . . . . . . . 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 7318	. . . . . . . . . . . 12 ⊢ ( [⊊] Or 𝑏 → [⊊] Or {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑐) ∈ 𝑏})
13	12	adantl 482	. . . . . . . . . . 11 ⊢ ((𝑏 ≠ ∅ ∧ [⊊] Or 𝑏) → [⊊] Or {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑐) ∈ 𝑏})
14	13	3ad2ant3 1128	. . . . . . . . . 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 3046	. . . . . . . . . . . . 13 ⊢ (𝑦 = {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑐) ∈ 𝑏} → (𝑦 ≠ ∅ ↔ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑐) ∈ 𝑏} ≠ ∅))
16		soeq2 5383	. . . . . . . . . . . . 13 ⊢ (𝑦 = {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑐) ∈ 𝑏} → ( [⊊] Or 𝑦 ↔ [⊊] Or {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑐) ∈ 𝑏}))
17	15, 16	anbi12d 630	. . . . . . . . . . . 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 4785	. . . . . . . . . . . . 13 ⊢ (𝑦 = {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑐) ∈ 𝑏} → ∩ 𝑦 = ∩ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑐) ∈ 𝑏})
19		id 22	. . . . . . . . . . . . 13 ⊢ (𝑦 = {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑐) ∈ 𝑏} → 𝑦 = {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑐) ∈ 𝑏})
20	18, 19	eleq12d 2877	. . . . . . . . . . . 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 346	. . . . . . . . . . 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 1130	. . . . . . . . . . 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 3977	. . . . . . . . . . . 12 ⊢ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑐) ∈ 𝑏} ⊆ 𝒫 𝐴
24		pwexg 5170	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ V)
25		elpw2g 5138	. . . . . . . . . . . . 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 18	. . . . . . . . . . . 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 259	. . . . . . . . . . 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 3565	. . . . . . . . . 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 695	. . . . . . . . 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 7317	. . . . . . . . . 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 17	. . . . . . . . 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 258	. . . . . . . 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 3985	. . . . . . . . 9 ⊢ (𝑚 = (𝐴 ∖ 𝑧) → (𝑚 ⊊ 𝑛 ↔ (𝐴 ∖ 𝑧) ⊊ 𝑛))
34		psseq1 3985	. . . . . . . . 9 ⊢ (𝑤 = (𝐴 ∖ 𝑛) → (𝑤 ⊊ 𝑧 ↔ (𝐴 ∖ 𝑛) ⊊ 𝑧))
35		pssdifcom1 4349	. . . . . . . . 9 ⊢ ((𝑧 ⊆ 𝐴 ∧ 𝑛 ⊆ 𝐴) → ((𝐴 ∖ 𝑧) ⊊ 𝑛 ↔ (𝐴 ∖ 𝑛) ⊊ 𝑧))
36	33, 34, 35	fin23lem11 9585	. . . . . . . 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 65	. . . . . . 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 1195	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑏 ⊆ 𝒫 𝐴) ∧ ∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [⊊] Or 𝑦) → ∩ 𝑦 ∈ 𝑦) ∧ (𝑏 ≠ ∅ ∧ [⊊] Or 𝑏)) → [⊊] Or 𝑏)
39		sorpssuni 7316	. . . . . . . 8 ⊢ ( [⊊] Or 𝑏 → (∃𝑚 ∈ 𝑏 ∀𝑛 ∈ 𝑏 ¬ 𝑚 ⊊ 𝑛 ↔ ∪ 𝑏 ∈ 𝑏))
40	38, 39	syl 17	. . . . . . 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 233	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑏 ⊆ 𝒫 𝐴) ∧ ∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [⊊] Or 𝑦) → ∩ 𝑦 ∈ 𝑦) ∧ (𝑏 ≠ ∅ ∧ [⊊] Or 𝑏)) → ∪ 𝑏 ∈ 𝑏)
42	41	3exp 1112	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑏 ⊆ 𝒫 𝐴) → (∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [⊊] Or 𝑦) → ∩ 𝑦 ∈ 𝑦) → ((𝑏 ≠ ∅ ∧ [⊊] Or 𝑏) → ∪ 𝑏 ∈ 𝑏)))
43	6, 42	sylan2 592	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑏 ∈ 𝒫 𝒫 𝐴) → (∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [⊊] Or 𝑦) → ∩ 𝑦 ∈ 𝑦) → ((𝑏 ≠ ∅ ∧ [⊊] Or 𝑏) → ∪ 𝑏 ∈ 𝑏)))
44	43	ralrimdva 3156	. . 3 ⊢ (𝐴 ∈ 𝑉 → (∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [⊊] Or 𝑦) → ∩ 𝑦 ∈ 𝑦) → ∀𝑏 ∈ 𝒫 𝒫 𝐴((𝑏 ≠ ∅ ∧ [⊊] Or 𝑏) → ∪ 𝑏 ∈ 𝑏)))
45		isfin2 9562	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ FinII ↔ ∀𝑏 ∈ 𝒫 𝒫 𝐴((𝑏 ≠ ∅ ∧ [⊊] Or 𝑏) → ∪ 𝑏 ∈ 𝑏)))
46	44, 45	sylibrd 260	. 2 ⊢ (𝐴 ∈ 𝑉 → (∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [⊊] Or 𝑦) → ∩ 𝑦 ∈ 𝑦) → 𝐴 ∈ FinII))
47	5, 46	impbid2 227	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ FinII ↔ ∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [⊊] Or 𝑦) → ∩ 𝑦 ∈ 𝑦)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 207   ∧ wa 396   ∧ w3a 1080   = wceq 1522   ∈ wcel 2081   ≠ wne 2984  ∀wral 3105  ∃wrex 3106  {crab 3109  Vcvv 3437   ∖ cdif 3856   ⊆ wss 3859   ⊊ wpss 3860  ∅c0 4211  𝒫 cpw 4453  ∪ cuni 4745  ∩ cint 4782   Or wor 5361   [⊊] crpss 7306  Fin^IIcfin2 9547

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221  ax-un 7319

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-ral 3110  df-rex 3111  df-rab 3114  df-v 3439  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-pss 3876  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-op 4479  df-uni 4746  df-int 4783  df-br 4963  df-opab 5025  df-po 5362  df-so 5363  df-xp 5449  df-rel 5450  df-rpss 7307  df-fin2 9554

This theorem is referenced by: (None)