Theorem finexdc 7097

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 2730	. . 3 ⊢ (𝑤 = ∅ → (∃𝑥 ∈ 𝑤 𝜑 ↔ ∃𝑥 ∈ ∅ 𝜑))
2	1	dcbid 845	. 2 ⊢ (𝑤 = ∅ → (DECID ∃𝑥 ∈ 𝑤 𝜑 ↔ DECID ∃𝑥 ∈ ∅ 𝜑))
3		rexeq 2730	. . 3 ⊢ (𝑤 = 𝑦 → (∃𝑥 ∈ 𝑤 𝜑 ↔ ∃𝑥 ∈ 𝑦 𝜑))
4	3	dcbid 845	. 2 ⊢ (𝑤 = 𝑦 → (DECID ∃𝑥 ∈ 𝑤 𝜑 ↔ DECID ∃𝑥 ∈ 𝑦 𝜑))
5		rexeq 2730	. . 3 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (∃𝑥 ∈ 𝑤 𝜑 ↔ ∃𝑥 ∈ (𝑦 ∪ {𝑧})𝜑))
6	5	dcbid 845	. 2 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (DECID ∃𝑥 ∈ 𝑤 𝜑 ↔ DECID ∃𝑥 ∈ (𝑦 ∪ {𝑧})𝜑))
7		rexeq 2730	. . 3 ⊢ (𝑤 = 𝐴 → (∃𝑥 ∈ 𝑤 𝜑 ↔ ∃𝑥 ∈ 𝐴 𝜑))
8	7	dcbid 845	. 2 ⊢ (𝑤 = 𝐴 → (DECID ∃𝑥 ∈ 𝑤 𝜑 ↔ DECID ∃𝑥 ∈ 𝐴 𝜑))
9		rex0 3511	. . . . 5 ⊢ ¬ ∃𝑥 ∈ ∅ 𝜑
10	9	olci 739	. . . 4 ⊢ (∃𝑥 ∈ ∅ 𝜑 ∨ ¬ ∃𝑥 ∈ ∅ 𝜑)
11		df-dc 842	. . . 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 3034	. . . . . . . . . 10 ⊢ ([𝑧 / 𝑥]𝜑 ↔ [𝑧 / 𝑥]𝜑)
16		rexsns 3709	. . . . . . . . . 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 741	. . . . . . 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 3386	. . . . . . 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 740	. . . . 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 842	. . . . 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 740	. . . . . . . 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 740	. . . . . 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 674	. . . . . . . . . . 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 759	. . . . . . . . 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 674	. . . . . . . 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 741	. . . . . 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 843	. . . . . 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 805	. . . 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 538	. . . . . . 7 ⊢ (((((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 DECID 𝜑) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ DECID ∃𝑥 ∈ 𝑦 𝜑) → 𝑧 ∈ (𝐴 ∖ 𝑦))
45	44	eldifad 3210	. . . . . 6 ⊢ (((((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 DECID 𝜑) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ DECID ∃𝑥 ∈ 𝑦 𝜑) → 𝑧 ∈ 𝐴)
46		simp-4r 544	. . . . . 6 ⊢ (((((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 DECID 𝜑) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ DECID ∃𝑥 ∈ 𝑦 𝜑) → ∀𝑥 ∈ 𝐴 DECID 𝜑)
47		nfs1v 1991	. . . . . . . 8 ⊢ Ⅎ𝑥[𝑧 / 𝑥]𝜑
48	47	nfdc 1706	. . . . . . 7 ⊢ Ⅎ𝑥DECID [𝑧 / 𝑥]𝜑
49		sbequ12 1818	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑))
50	49	dcbid 845	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (DECID 𝜑 ↔ DECID [𝑧 / 𝑥]𝜑))
51	48, 50	rspc 2903	. . . . . 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 843	. . . . 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 805	. . 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 7086	1 ⊢ ((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 DECID 𝜑) → DECID ∃𝑥 ∈ 𝐴 𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ∨ wo 715  DECID wdc 841   = wceq 1397  [wsb 1809   ∈ wcel 2201  ∀wral 2509  ∃wrex 2510  [wsbc 3030   ∖ cdif 3196   ∪ cun 3197   ⊆ wss 3199  ∅c0 3493  {csn 3670  Fincfn 6914

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2203  ax-14 2204  ax-ext 2212  ax-coll 4205  ax-sep 4208  ax-nul 4216  ax-pow 4266  ax-pr 4301  ax-un 4532  ax-setind 4637  ax-iinf 4688

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1810  df-eu 2081  df-mo 2082  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-ne 2402  df-ral 2514  df-rex 2515  df-reu 2516  df-rab 2518  df-v 2803  df-sbc 3031  df-csb 3127  df-dif 3201  df-un 3203  df-in 3205  df-ss 3212  df-nul 3494  df-if 3605  df-pw 3655  df-sn 3676  df-pr 3677  df-op 3679  df-uni 3895  df-int 3930  df-iun 3973  df-br 4090  df-opab 4152  df-mpt 4153  df-tr 4189  df-id 4392  df-iord 4465  df-on 4467  df-suc 4470  df-iom 4691  df-xp 4733  df-rel 4734  df-cnv 4735  df-co 4736  df-dm 4737  df-rn 4738  df-res 4739  df-ima 4740  df-iota 5288  df-fun 5330  df-fn 5331  df-f 5332  df-f1 5333  df-fo 5334  df-f1o 5335  df-fv 5336  df-er 6707  df-en 6915  df-fin 6917

This theorem is referenced by:  dfrex2fin  7098  nninfwlpoimlemg  7379  nninfwlpoimlemginf  7380  4sqleminfi  12993