Theorem finexdc 6963

Description: Decidability of existence, over a finite set and defined by a decidable proposition. (Contributed by Jim Kingdon, 12-Jul-2022.)

Assertion

Ref	Expression
finexdc	⊢ ((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 DECID 𝜑) → DECID ∃𝑥 ∈ 𝐴 𝜑)

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem finexdc

Dummy variables 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		rexeq 2694	. . 3 ⊢ (𝑤 = ∅ → (∃𝑥 ∈ 𝑤 𝜑 ↔ ∃𝑥 ∈ ∅ 𝜑))
2	1	dcbid 839	. 2 ⊢ (𝑤 = ∅ → (DECID ∃𝑥 ∈ 𝑤 𝜑 ↔ DECID ∃𝑥 ∈ ∅ 𝜑))
3		rexeq 2694	. . 3 ⊢ (𝑤 = 𝑦 → (∃𝑥 ∈ 𝑤 𝜑 ↔ ∃𝑥 ∈ 𝑦 𝜑))
4	3	dcbid 839	. 2 ⊢ (𝑤 = 𝑦 → (DECID ∃𝑥 ∈ 𝑤 𝜑 ↔ DECID ∃𝑥 ∈ 𝑦 𝜑))
5		rexeq 2694	. . 3 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (∃𝑥 ∈ 𝑤 𝜑 ↔ ∃𝑥 ∈ (𝑦 ∪ {𝑧})𝜑))
6	5	dcbid 839	. 2 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (DECID ∃𝑥 ∈ 𝑤 𝜑 ↔ DECID ∃𝑥 ∈ (𝑦 ∪ {𝑧})𝜑))
7		rexeq 2694	. . 3 ⊢ (𝑤 = 𝐴 → (∃𝑥 ∈ 𝑤 𝜑 ↔ ∃𝑥 ∈ 𝐴 𝜑))
8	7	dcbid 839	. 2 ⊢ (𝑤 = 𝐴 → (DECID ∃𝑥 ∈ 𝑤 𝜑 ↔ DECID ∃𝑥 ∈ 𝐴 𝜑))
9		rex0 3468	. . . . 5 ⊢ ¬ ∃𝑥 ∈ ∅ 𝜑
10	9	olci 733	. . . 4 ⊢ (∃𝑥 ∈ ∅ 𝜑 ∨ ¬ ∃𝑥 ∈ ∅ 𝜑)
11		df-dc 836	. . . 4 ⊢ (DECID ∃𝑥 ∈ ∅ 𝜑 ↔ (∃𝑥 ∈ ∅ 𝜑 ∨ ¬ ∃𝑥 ∈ ∅ 𝜑))
12	10, 11	mpbir 146	. . 3 ⊢ DECID ∃𝑥 ∈ ∅ 𝜑
13	12	a1i 9	. 2 ⊢ ((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 DECID 𝜑) → DECID ∃𝑥 ∈ ∅ 𝜑)
14		simpr 110	. . . . . . . . 9 ⊢ ((((((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 DECID 𝜑) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ DECID ∃𝑥 ∈ 𝑦 𝜑) ∧ [𝑧 / 𝑥]𝜑) → [𝑧 / 𝑥]𝜑)
15		sbsbc 2993	. . . . . . . . . 10 ⊢ ([𝑧 / 𝑥]𝜑 ↔ [𝑧 / 𝑥]𝜑)
16		rexsns 3661	. . . . . . . . . 10 ⊢ (∃𝑥 ∈ {𝑧}𝜑 ↔ [𝑧 / 𝑥]𝜑)
17	15, 16	bitr4i 187	. . . . . . . . 9 ⊢ ([𝑧 / 𝑥]𝜑 ↔ ∃𝑥 ∈ {𝑧}𝜑)
18	14, 17	sylib 122	. . . . . . . 8 ⊢ ((((((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 DECID 𝜑) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ DECID ∃𝑥 ∈ 𝑦 𝜑) ∧ [𝑧 / 𝑥]𝜑) → ∃𝑥 ∈ {𝑧}𝜑)
19	18	olcd 735	. . . . . . 7 ⊢ ((((((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 DECID 𝜑) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ DECID ∃𝑥 ∈ 𝑦 𝜑) ∧ [𝑧 / 𝑥]𝜑) → (∃𝑥 ∈ 𝑦 𝜑 ∨ ∃𝑥 ∈ {𝑧}𝜑))
20		rexun 3343	. . . . . . 7 ⊢ (∃𝑥 ∈ (𝑦 ∪ {𝑧})𝜑 ↔ (∃𝑥 ∈ 𝑦 𝜑 ∨ ∃𝑥 ∈ {𝑧}𝜑))
21	19, 20	sylibr 134	. . . . . 6 ⊢ ((((((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 DECID 𝜑) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ DECID ∃𝑥 ∈ 𝑦 𝜑) ∧ [𝑧 / 𝑥]𝜑) → ∃𝑥 ∈ (𝑦 ∪ {𝑧})𝜑)
22	21	orcd 734	. . . . 5 ⊢ ((((((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 DECID 𝜑) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ DECID ∃𝑥 ∈ 𝑦 𝜑) ∧ [𝑧 / 𝑥]𝜑) → (∃𝑥 ∈ (𝑦 ∪ {𝑧})𝜑 ∨ ¬ ∃𝑥 ∈ (𝑦 ∪ {𝑧})𝜑))
23		df-dc 836	. . . . 5 ⊢ (DECID ∃𝑥 ∈ (𝑦 ∪ {𝑧})𝜑 ↔ (∃𝑥 ∈ (𝑦 ∪ {𝑧})𝜑 ∨ ¬ ∃𝑥 ∈ (𝑦 ∪ {𝑧})𝜑))
24	22, 23	sylibr 134	. . . 4 ⊢ ((((((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 DECID 𝜑) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ DECID ∃𝑥 ∈ 𝑦 𝜑) ∧ [𝑧 / 𝑥]𝜑) → DECID ∃𝑥 ∈ (𝑦 ∪ {𝑧})𝜑)
25		simpr 110	. . . . . . . . 9 ⊢ (((((((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 DECID 𝜑) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ DECID ∃𝑥 ∈ 𝑦 𝜑) ∧ ¬ [𝑧 / 𝑥]𝜑) ∧ ∃𝑥 ∈ 𝑦 𝜑) → ∃𝑥 ∈ 𝑦 𝜑)
26	25	orcd 734	. . . . . . . 8 ⊢ (((((((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 DECID 𝜑) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ DECID ∃𝑥 ∈ 𝑦 𝜑) ∧ ¬ [𝑧 / 𝑥]𝜑) ∧ ∃𝑥 ∈ 𝑦 𝜑) → (∃𝑥 ∈ 𝑦 𝜑 ∨ ∃𝑥 ∈ {𝑧}𝜑))
27	26, 20	sylibr 134	. . . . . . 7 ⊢ (((((((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 DECID 𝜑) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ DECID ∃𝑥 ∈ 𝑦 𝜑) ∧ ¬ [𝑧 / 𝑥]𝜑) ∧ ∃𝑥 ∈ 𝑦 𝜑) → ∃𝑥 ∈ (𝑦 ∪ {𝑧})𝜑)
28	27	orcd 734	. . . . . 6 ⊢ (((((((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 DECID 𝜑) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ DECID ∃𝑥 ∈ 𝑦 𝜑) ∧ ¬ [𝑧 / 𝑥]𝜑) ∧ ∃𝑥 ∈ 𝑦 𝜑) → (∃𝑥 ∈ (𝑦 ∪ {𝑧})𝜑 ∨ ¬ ∃𝑥 ∈ (𝑦 ∪ {𝑧})𝜑))
29	28, 23	sylibr 134	. . . . 5 ⊢ (((((((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 DECID 𝜑) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ DECID ∃𝑥 ∈ 𝑦 𝜑) ∧ ¬ [𝑧 / 𝑥]𝜑) ∧ ∃𝑥 ∈ 𝑦 𝜑) → DECID ∃𝑥 ∈ (𝑦 ∪ {𝑧})𝜑)
30		simpr 110	. . . . . . . . 9 ⊢ (((((((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 DECID 𝜑) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ DECID ∃𝑥 ∈ 𝑦 𝜑) ∧ ¬ [𝑧 / 𝑥]𝜑) ∧ ¬ ∃𝑥 ∈ 𝑦 𝜑) → ¬ ∃𝑥 ∈ 𝑦 𝜑)
31		simpr 110	. . . . . . . . . . 11 ⊢ ((((((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 DECID 𝜑) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ DECID ∃𝑥 ∈ 𝑦 𝜑) ∧ ¬ [𝑧 / 𝑥]𝜑) → ¬ [𝑧 / 𝑥]𝜑)
32	17	notbii 669	. . . . . . . . . . 11 ⊢ (¬ [𝑧 / 𝑥]𝜑 ↔ ¬ ∃𝑥 ∈ {𝑧}𝜑)
33	31, 32	sylib 122	. . . . . . . . . 10 ⊢ ((((((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 DECID 𝜑) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ DECID ∃𝑥 ∈ 𝑦 𝜑) ∧ ¬ [𝑧 / 𝑥]𝜑) → ¬ ∃𝑥 ∈ {𝑧}𝜑)
34	33	adantr 276	. . . . . . . . 9 ⊢ (((((((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 DECID 𝜑) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ DECID ∃𝑥 ∈ 𝑦 𝜑) ∧ ¬ [𝑧 / 𝑥]𝜑) ∧ ¬ ∃𝑥 ∈ 𝑦 𝜑) → ¬ ∃𝑥 ∈ {𝑧}𝜑)
35		ioran 753	. . . . . . . . 9 ⊢ (¬ (∃𝑥 ∈ 𝑦 𝜑 ∨ ∃𝑥 ∈ {𝑧}𝜑) ↔ (¬ ∃𝑥 ∈ 𝑦 𝜑 ∧ ¬ ∃𝑥 ∈ {𝑧}𝜑))
36	30, 34, 35	sylanbrc 417	. . . . . . . 8 ⊢ (((((((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 DECID 𝜑) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ DECID ∃𝑥 ∈ 𝑦 𝜑) ∧ ¬ [𝑧 / 𝑥]𝜑) ∧ ¬ ∃𝑥 ∈ 𝑦 𝜑) → ¬ (∃𝑥 ∈ 𝑦 𝜑 ∨ ∃𝑥 ∈ {𝑧}𝜑))
37	20	notbii 669	. . . . . . . 8 ⊢ (¬ ∃𝑥 ∈ (𝑦 ∪ {𝑧})𝜑 ↔ ¬ (∃𝑥 ∈ 𝑦 𝜑 ∨ ∃𝑥 ∈ {𝑧}𝜑))
38	36, 37	sylibr 134	. . . . . . 7 ⊢ (((((((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 DECID 𝜑) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ DECID ∃𝑥 ∈ 𝑦 𝜑) ∧ ¬ [𝑧 / 𝑥]𝜑) ∧ ¬ ∃𝑥 ∈ 𝑦 𝜑) → ¬ ∃𝑥 ∈ (𝑦 ∪ {𝑧})𝜑)
39	38	olcd 735	. . . . . 6 ⊢ (((((((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 DECID 𝜑) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ DECID ∃𝑥 ∈ 𝑦 𝜑) ∧ ¬ [𝑧 / 𝑥]𝜑) ∧ ¬ ∃𝑥 ∈ 𝑦 𝜑) → (∃𝑥 ∈ (𝑦 ∪ {𝑧})𝜑 ∨ ¬ ∃𝑥 ∈ (𝑦 ∪ {𝑧})𝜑))
40	39, 23	sylibr 134	. . . . 5 ⊢ (((((((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 DECID 𝜑) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ DECID ∃𝑥 ∈ 𝑦 𝜑) ∧ ¬ [𝑧 / 𝑥]𝜑) ∧ ¬ ∃𝑥 ∈ 𝑦 𝜑) → DECID ∃𝑥 ∈ (𝑦 ∪ {𝑧})𝜑)
41		exmiddc 837	. . . . . 6 ⊢ (DECID ∃𝑥 ∈ 𝑦 𝜑 → (∃𝑥 ∈ 𝑦 𝜑 ∨ ¬ ∃𝑥 ∈ 𝑦 𝜑))
42	41	ad2antlr 489	. . . . 5 ⊢ ((((((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 DECID 𝜑) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ DECID ∃𝑥 ∈ 𝑦 𝜑) ∧ ¬ [𝑧 / 𝑥]𝜑) → (∃𝑥 ∈ 𝑦 𝜑 ∨ ¬ ∃𝑥 ∈ 𝑦 𝜑))
43	29, 40, 42	mpjaodan 799	. . . 4 ⊢ ((((((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 DECID 𝜑) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ DECID ∃𝑥 ∈ 𝑦 𝜑) ∧ ¬ [𝑧 / 𝑥]𝜑) → DECID ∃𝑥 ∈ (𝑦 ∪ {𝑧})𝜑)
44		simplrr 536	. . . . . . 7 ⊢ (((((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 DECID 𝜑) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ DECID ∃𝑥 ∈ 𝑦 𝜑) → 𝑧 ∈ (𝐴 ∖ 𝑦))
45	44	eldifad 3168	. . . . . 6 ⊢ (((((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 DECID 𝜑) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ DECID ∃𝑥 ∈ 𝑦 𝜑) → 𝑧 ∈ 𝐴)
46		simp-4r 542	. . . . . 6 ⊢ (((((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 DECID 𝜑) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ DECID ∃𝑥 ∈ 𝑦 𝜑) → ∀𝑥 ∈ 𝐴 DECID 𝜑)
47		nfs1v 1958	. . . . . . . 8 ⊢ Ⅎ𝑥[𝑧 / 𝑥]𝜑
48	47	nfdc 1673	. . . . . . 7 ⊢ Ⅎ𝑥DECID [𝑧 / 𝑥]𝜑
49		sbequ12 1785	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑))
50	49	dcbid 839	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (DECID 𝜑 ↔ DECID [𝑧 / 𝑥]𝜑))
51	48, 50	rspc 2862	. . . . . 6 ⊢ (𝑧 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 DECID 𝜑 → DECID [𝑧 / 𝑥]𝜑))
52	45, 46, 51	sylc 62	. . . . 5 ⊢ (((((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 DECID 𝜑) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ DECID ∃𝑥 ∈ 𝑦 𝜑) → DECID [𝑧 / 𝑥]𝜑)
53		exmiddc 837	. . . . 5 ⊢ (DECID [𝑧 / 𝑥]𝜑 → ([𝑧 / 𝑥]𝜑 ∨ ¬ [𝑧 / 𝑥]𝜑))
54	52, 53	syl 14	. . . 4 ⊢ (((((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 DECID 𝜑) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ DECID ∃𝑥 ∈ 𝑦 𝜑) → ([𝑧 / 𝑥]𝜑 ∨ ¬ [𝑧 / 𝑥]𝜑))
55	24, 43, 54	mpjaodan 799	. . 3 ⊢ (((((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 DECID 𝜑) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ DECID ∃𝑥 ∈ 𝑦 𝜑) → DECID ∃𝑥 ∈ (𝑦 ∪ {𝑧})𝜑)
56	55	ex 115	. 2 ⊢ ((((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 DECID 𝜑) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → (DECID ∃𝑥 ∈ 𝑦 𝜑 → DECID ∃𝑥 ∈ (𝑦 ∪ {𝑧})𝜑))
57		simpl 109	. 2 ⊢ ((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 DECID 𝜑) → 𝐴 ∈ Fin)
58	2, 4, 6, 8, 13, 56, 57	findcard2sd 6953	1 ⊢ ((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 DECID 𝜑) → DECID ∃𝑥 ∈ 𝐴 𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ∨ wo 709  DECID wdc 835   = wceq 1364  [wsb 1776   ∈ wcel 2167  ∀wral 2475  ∃wrex 2476  [wsbc 2989   ∖ cdif 3154   ∪ cun 3155   ⊆ wss 3157  ∅c0 3450  {csn 3622  Fincfn 6799

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-iinf 4624

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-if 3562  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-tr 4132  df-id 4328  df-iord 4401  df-on 4403  df-suc 4406  df-iom 4627  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-er 6592  df-en 6800  df-fin 6802

This theorem is referenced by:  dfrex2fin  6964  nninfwlpoimlemg  7241  nninfwlpoimlemginf  7242  4sqleminfi  12566