Theorem isfin2-2 10424

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 4567	. . . 4 ⊢ (𝑦 ∈ 𝒫 𝒫 𝐴 → 𝑦 ⊆ 𝒫 𝐴)
2		fin2i2 10423	. . . . 5 ⊢ (((𝐴 ∈ FinII ∧ 𝑦 ⊆ 𝒫 𝐴) ∧ (𝑦 ≠ ∅ ∧ [⊊] Or 𝑦)) → ∩ 𝑦 ∈ 𝑦)
3	2	ex 418	. . . 4 ⊢ ((𝐴 ∈ FinII ∧ 𝑦 ⊆ 𝒫 𝐴) → ((𝑦 ≠ ∅ ∧ [⊊] Or 𝑦) → ∩ 𝑦 ∈ 𝑦))
4	1, 3	sylan2 605	. . 3 ⊢ ((𝐴 ∈ FinII ∧ 𝑦 ∈ 𝒫 𝒫 𝐴) → ((𝑦 ≠ ∅ ∧ [⊊] Or 𝑦) → ∩ 𝑦 ∈ 𝑦))
5	4	ralrimiva 3158	. 2 ⊢ (𝐴 ∈ FinII → ∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [⊊] Or 𝑦) → ∩ 𝑦 ∈ 𝑦))
6		elpwi 4567	. . . . 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 10421	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑏 ⊆ 𝒫 𝐴 ∧ 𝑏 ≠ ∅) → {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑐) ∈ 𝑏} ≠ ∅)
11	8, 7, 9, 10	syl3anc 1399	. . . . . . . . . 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 7774	. . . . . . . . . . . 12 ⊢ ( [⊊] Or 𝑏 → [⊊] Or {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑐) ∈ 𝑏})
13	12	adantl 487	. . . . . . . . . . 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 3021	. . . . . . . . . . . . 13 ⊢ (𝑦 = {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑐) ∈ 𝑏} → (𝑦 ≠ ∅ ↔ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑐) ∈ 𝑏} ≠ ∅))
16		soeq2 5585	. . . . . . . . . . . . 13 ⊢ (𝑦 = {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑐) ∈ 𝑏} → ( [⊊] Or 𝑦 ↔ [⊊] Or {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑐) ∈ 𝑏}))
17	15, 16	anbi12d 644	. . . . . . . . . . . 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 4913	. . . . . . . . . . . . 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 2858	. . . . . . . . . . . 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 4008	. . . . . . . . . . . 12 ⊢ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑐) ∈ 𝑏} ⊆ 𝒫 𝐴
24		pwexg 5344	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ V)
25		elpw2g 5299	. . . . . . . . . . . . 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 3581	. . . . . . . . . 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 712	. . . . . . . . 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 7773	. . . . . . . . . 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 4020	. . . . . . . . 9 ⊢ (𝑚 = (𝐴 ∖ 𝑧) → (𝑚 ⊊ 𝑛 ↔ (𝐴 ∖ 𝑧) ⊊ 𝑛))
34		psseq1 4020	. . . . . . . . 9 ⊢ (𝑤 = (𝐴 ∖ 𝑛) → (𝑤 ⊊ 𝑧 ↔ (𝐴 ∖ 𝑛) ⊊ 𝑧))
35		pssdifcom1 4448	. . . . . . . . 9 ⊢ ((𝑧 ⊆ 𝐴 ∧ 𝑛 ⊆ 𝐴) → ((𝐴 ∖ 𝑧) ⊊ 𝑛 ↔ (𝐴 ∖ 𝑛) ⊊ 𝑧))
36	33, 34, 35	fin23lem11 10422	. . . . . . . 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 7772	. . . . . . . 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 605	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑏 ∈ 𝒫 𝒫 𝐴) → (∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [⊊] Or 𝑦) → ∩ 𝑦 ∈ 𝑦) → ((𝑏 ≠ ∅ ∧ [⊊] Or 𝑏) → ∪ 𝑏 ∈ 𝑏)))
44	43	ralrimdva 3166	. . 3 ⊢ (𝐴 ∈ 𝑉 → (∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [⊊] Or 𝑦) → ∩ 𝑦 ∈ 𝑦) → ∀𝑏 ∈ 𝒫 𝒫 𝐴((𝑏 ≠ ∅ ∧ [⊊] Or 𝑏) → ∪ 𝑏 ∈ 𝑏)))
45		isfin2 10399	. . 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)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147   ≠ wne 2959  ∀wral 3080  ∃wrex 3090  {crab 3416  Vcvv 3454   ⊆ wss 3900   ⊊ wpss 4016   ∖ cdif 4048  ∅c0 4281  𝒫 cpw 4560  ∪ cuni 4870  ∩ cint 4910   Or wor 5562   [⊊] crpss 7762  Fin^IIcfin2 10384

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-pow 5331  ax-pr 5395  ax-un 7775

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-int 4911  df-br 5107  df-opab 5171  df-po 5563  df-so 5564  df-xp 5661  df-rel 5662  df-rpss 7763  df-fin2 10391

This theorem is used by: (None)