Theorem finexdc 6902

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 2674	. . 3 ⊢ (𝑤 = ∅ → (∃𝑥 ∈ 𝑤 𝜑 ↔ ∃𝑥 ∈ ∅ 𝜑))
2	1	dcbid 838	. 2 ⊢ (𝑤 = ∅ → (DECID ∃𝑥 ∈ 𝑤 𝜑 ↔ DECID ∃𝑥 ∈ ∅ 𝜑))
3		rexeq 2674	. . 3 ⊢ (𝑤 = 𝑦 → (∃𝑥 ∈ 𝑤 𝜑 ↔ ∃𝑥 ∈ 𝑦 𝜑))
4	3	dcbid 838	. 2 ⊢ (𝑤 = 𝑦 → (DECID ∃𝑥 ∈ 𝑤 𝜑 ↔ DECID ∃𝑥 ∈ 𝑦 𝜑))
5		rexeq 2674	. . 3 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (∃𝑥 ∈ 𝑤 𝜑 ↔ ∃𝑥 ∈ (𝑦 ∪ {𝑧})𝜑))
6	5	dcbid 838	. 2 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (DECID ∃𝑥 ∈ 𝑤 𝜑 ↔ DECID ∃𝑥 ∈ (𝑦 ∪ {𝑧})𝜑))
7		rexeq 2674	. . 3 ⊢ (𝑤 = 𝐴 → (∃𝑥 ∈ 𝑤 𝜑 ↔ ∃𝑥 ∈ 𝐴 𝜑))
8	7	dcbid 838	. 2 ⊢ (𝑤 = 𝐴 → (DECID ∃𝑥 ∈ 𝑤 𝜑 ↔ DECID ∃𝑥 ∈ 𝐴 𝜑))
9		rex0 3441	. . . . 5 ⊢ ¬ ∃𝑥 ∈ ∅ 𝜑
10	9	olci 732	. . . 4 ⊢ (∃𝑥 ∈ ∅ 𝜑 ∨ ¬ ∃𝑥 ∈ ∅ 𝜑)
11		df-dc 835	. . . 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 2967	. . . . . . . . . 10 ⊢ ([𝑧 / 𝑥]𝜑 ↔ [𝑧 / 𝑥]𝜑)
16		rexsns 3632	. . . . . . . . . 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 734	. . . . . . 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 3316	. . . . . . 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 733	. . . . 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 835	. . . . 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 733	. . . . . . . 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 733	. . . . . 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 668	. . . . . . . . . . 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 752	. . . . . . . . 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 668	. . . . . . . 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 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))
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 836	. . . . . 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 798	. . . 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 3141	. . . . . 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 1939	. . . . . . . 8 ⊢ Ⅎ𝑥[𝑧 / 𝑥]𝜑
48	47	nfdc 1659	. . . . . . 7 ⊢ Ⅎ𝑥DECID [𝑧 / 𝑥]𝜑
49		sbequ12 1771	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑))
50	49	dcbid 838	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (DECID 𝜑 ↔ DECID [𝑧 / 𝑥]𝜑))
51	48, 50	rspc 2836	. . . . . 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 836	. . . . 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 798	. . 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 6892	1 ⊢ ((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 DECID 𝜑) → DECID ∃𝑥 ∈ 𝐴 𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ∨ wo 708  DECID wdc 834   = wceq 1353  [wsb 1762   ∈ wcel 2148  ∀wral 2455  ∃wrex 2456  [wsbc 2963   ∖ cdif 3127   ∪ cun 3128   ⊆ wss 3130  ∅c0 3423  {csn 3593  Fincfn 6740

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4119  ax-sep 4122  ax-nul 4130  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537  ax-iinf 4588

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2740  df-sbc 2964  df-csb 3059  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-if 3536  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-int 3846  df-iun 3889  df-br 4005  df-opab 4066  df-mpt 4067  df-tr 4103  df-id 4294  df-iord 4367  df-on 4369  df-suc 4372  df-iom 4591  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225  df-er 6535  df-en 6741  df-fin 6743

This theorem is referenced by:  dfrex2fin  6903  nninfwlpoimlemg  7173  nninfwlpoimlemginf  7174