Theorem nneob 7732

Description: A natural number is even iff its successor is odd. (Contributed by NM, 26-Jan-2006.) (Revised by Mario Carneiro, 15-Nov-2014.)

Assertion

Ref	Expression
nneob	⊢ (𝐴 ∈ ω → (∃𝑥 ∈ ω 𝐴 = (2𝑜 ·𝑜 𝑥) ↔ ¬ ∃𝑥 ∈ ω suc 𝐴 = (2𝑜 ·𝑜 𝑥)))

Distinct variable group:   𝑥,𝐴

Proof of Theorem nneob

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

Step	Hyp	Ref	Expression
1		oveq2 6658	. . . . 5 ⊢ (𝑥 = 𝑦 → (2𝑜 ·𝑜 𝑥) = (2𝑜 ·𝑜 𝑦))
2	1	eqeq2d 2632	. . . 4 ⊢ (𝑥 = 𝑦 → (𝐴 = (2𝑜 ·𝑜 𝑥) ↔ 𝐴 = (2𝑜 ·𝑜 𝑦)))
3	2	cbvrexv 3172	. . 3 ⊢ (∃𝑥 ∈ ω 𝐴 = (2𝑜 ·𝑜 𝑥) ↔ ∃𝑦 ∈ ω 𝐴 = (2𝑜 ·𝑜 𝑦))
4		nnneo 7731	. . . . . . 7 ⊢ ((𝑦 ∈ ω ∧ 𝑥 ∈ ω ∧ 𝐴 = (2𝑜 ·𝑜 𝑦)) → ¬ suc 𝐴 = (2𝑜 ·𝑜 𝑥))
5	4	3com23 1271	. . . . . 6 ⊢ ((𝑦 ∈ ω ∧ 𝐴 = (2𝑜 ·𝑜 𝑦) ∧ 𝑥 ∈ ω) → ¬ suc 𝐴 = (2𝑜 ·𝑜 𝑥))
6	5	3expa 1265	. . . . 5 ⊢ (((𝑦 ∈ ω ∧ 𝐴 = (2𝑜 ·𝑜 𝑦)) ∧ 𝑥 ∈ ω) → ¬ suc 𝐴 = (2𝑜 ·𝑜 𝑥))
7	6	nrexdv 3001	. . . 4 ⊢ ((𝑦 ∈ ω ∧ 𝐴 = (2𝑜 ·𝑜 𝑦)) → ¬ ∃𝑥 ∈ ω suc 𝐴 = (2𝑜 ·𝑜 𝑥))
8	7	rexlimiva 3028	. . 3 ⊢ (∃𝑦 ∈ ω 𝐴 = (2𝑜 ·𝑜 𝑦) → ¬ ∃𝑥 ∈ ω suc 𝐴 = (2𝑜 ·𝑜 𝑥))
9	3, 8	sylbi 207	. 2 ⊢ (∃𝑥 ∈ ω 𝐴 = (2𝑜 ·𝑜 𝑥) → ¬ ∃𝑥 ∈ ω suc 𝐴 = (2𝑜 ·𝑜 𝑥))
10		suceq 5790	. . . . . . 7 ⊢ (𝑦 = ∅ → suc 𝑦 = suc ∅)
11	10	eqeq1d 2624	. . . . . 6 ⊢ (𝑦 = ∅ → (suc 𝑦 = (2𝑜 ·𝑜 𝑥) ↔ suc ∅ = (2𝑜 ·𝑜 𝑥)))
12	11	rexbidv 3052	. . . . 5 ⊢ (𝑦 = ∅ → (∃𝑥 ∈ ω suc 𝑦 = (2𝑜 ·𝑜 𝑥) ↔ ∃𝑥 ∈ ω suc ∅ = (2𝑜 ·𝑜 𝑥)))
13	12	notbid 308	. . . 4 ⊢ (𝑦 = ∅ → (¬ ∃𝑥 ∈ ω suc 𝑦 = (2𝑜 ·𝑜 𝑥) ↔ ¬ ∃𝑥 ∈ ω suc ∅ = (2𝑜 ·𝑜 𝑥)))
14		eqeq1 2626	. . . . 5 ⊢ (𝑦 = ∅ → (𝑦 = (2𝑜 ·𝑜 𝑥) ↔ ∅ = (2𝑜 ·𝑜 𝑥)))
15	14	rexbidv 3052	. . . 4 ⊢ (𝑦 = ∅ → (∃𝑥 ∈ ω 𝑦 = (2𝑜 ·𝑜 𝑥) ↔ ∃𝑥 ∈ ω ∅ = (2𝑜 ·𝑜 𝑥)))
16	13, 15	imbi12d 334	. . 3 ⊢ (𝑦 = ∅ → ((¬ ∃𝑥 ∈ ω suc 𝑦 = (2𝑜 ·𝑜 𝑥) → ∃𝑥 ∈ ω 𝑦 = (2𝑜 ·𝑜 𝑥)) ↔ (¬ ∃𝑥 ∈ ω suc ∅ = (2𝑜 ·𝑜 𝑥) → ∃𝑥 ∈ ω ∅ = (2𝑜 ·𝑜 𝑥))))
17		suceq 5790	. . . . . . 7 ⊢ (𝑦 = 𝑧 → suc 𝑦 = suc 𝑧)
18	17	eqeq1d 2624	. . . . . 6 ⊢ (𝑦 = 𝑧 → (suc 𝑦 = (2𝑜 ·𝑜 𝑥) ↔ suc 𝑧 = (2𝑜 ·𝑜 𝑥)))
19	18	rexbidv 3052	. . . . 5 ⊢ (𝑦 = 𝑧 → (∃𝑥 ∈ ω suc 𝑦 = (2𝑜 ·𝑜 𝑥) ↔ ∃𝑥 ∈ ω suc 𝑧 = (2𝑜 ·𝑜 𝑥)))
20	19	notbid 308	. . . 4 ⊢ (𝑦 = 𝑧 → (¬ ∃𝑥 ∈ ω suc 𝑦 = (2𝑜 ·𝑜 𝑥) ↔ ¬ ∃𝑥 ∈ ω suc 𝑧 = (2𝑜 ·𝑜 𝑥)))
21		eqeq1 2626	. . . . 5 ⊢ (𝑦 = 𝑧 → (𝑦 = (2𝑜 ·𝑜 𝑥) ↔ 𝑧 = (2𝑜 ·𝑜 𝑥)))
22	21	rexbidv 3052	. . . 4 ⊢ (𝑦 = 𝑧 → (∃𝑥 ∈ ω 𝑦 = (2𝑜 ·𝑜 𝑥) ↔ ∃𝑥 ∈ ω 𝑧 = (2𝑜 ·𝑜 𝑥)))
23	20, 22	imbi12d 334	. . 3 ⊢ (𝑦 = 𝑧 → ((¬ ∃𝑥 ∈ ω suc 𝑦 = (2𝑜 ·𝑜 𝑥) → ∃𝑥 ∈ ω 𝑦 = (2𝑜 ·𝑜 𝑥)) ↔ (¬ ∃𝑥 ∈ ω suc 𝑧 = (2𝑜 ·𝑜 𝑥) → ∃𝑥 ∈ ω 𝑧 = (2𝑜 ·𝑜 𝑥))))
24		suceq 5790	. . . . . . 7 ⊢ (𝑦 = suc 𝑧 → suc 𝑦 = suc suc 𝑧)
25	24	eqeq1d 2624	. . . . . 6 ⊢ (𝑦 = suc 𝑧 → (suc 𝑦 = (2𝑜 ·𝑜 𝑥) ↔ suc suc 𝑧 = (2𝑜 ·𝑜 𝑥)))
26	25	rexbidv 3052	. . . . 5 ⊢ (𝑦 = suc 𝑧 → (∃𝑥 ∈ ω suc 𝑦 = (2𝑜 ·𝑜 𝑥) ↔ ∃𝑥 ∈ ω suc suc 𝑧 = (2𝑜 ·𝑜 𝑥)))
27	26	notbid 308	. . . 4 ⊢ (𝑦 = suc 𝑧 → (¬ ∃𝑥 ∈ ω suc 𝑦 = (2𝑜 ·𝑜 𝑥) ↔ ¬ ∃𝑥 ∈ ω suc suc 𝑧 = (2𝑜 ·𝑜 𝑥)))
28		eqeq1 2626	. . . . 5 ⊢ (𝑦 = suc 𝑧 → (𝑦 = (2𝑜 ·𝑜 𝑥) ↔ suc 𝑧 = (2𝑜 ·𝑜 𝑥)))
29	28	rexbidv 3052	. . . 4 ⊢ (𝑦 = suc 𝑧 → (∃𝑥 ∈ ω 𝑦 = (2𝑜 ·𝑜 𝑥) ↔ ∃𝑥 ∈ ω suc 𝑧 = (2𝑜 ·𝑜 𝑥)))
30	27, 29	imbi12d 334	. . 3 ⊢ (𝑦 = suc 𝑧 → ((¬ ∃𝑥 ∈ ω suc 𝑦 = (2𝑜 ·𝑜 𝑥) → ∃𝑥 ∈ ω 𝑦 = (2𝑜 ·𝑜 𝑥)) ↔ (¬ ∃𝑥 ∈ ω suc suc 𝑧 = (2𝑜 ·𝑜 𝑥) → ∃𝑥 ∈ ω suc 𝑧 = (2𝑜 ·𝑜 𝑥))))
31		suceq 5790	. . . . . . 7 ⊢ (𝑦 = 𝐴 → suc 𝑦 = suc 𝐴)
32	31	eqeq1d 2624	. . . . . 6 ⊢ (𝑦 = 𝐴 → (suc 𝑦 = (2𝑜 ·𝑜 𝑥) ↔ suc 𝐴 = (2𝑜 ·𝑜 𝑥)))
33	32	rexbidv 3052	. . . . 5 ⊢ (𝑦 = 𝐴 → (∃𝑥 ∈ ω suc 𝑦 = (2𝑜 ·𝑜 𝑥) ↔ ∃𝑥 ∈ ω suc 𝐴 = (2𝑜 ·𝑜 𝑥)))
34	33	notbid 308	. . . 4 ⊢ (𝑦 = 𝐴 → (¬ ∃𝑥 ∈ ω suc 𝑦 = (2𝑜 ·𝑜 𝑥) ↔ ¬ ∃𝑥 ∈ ω suc 𝐴 = (2𝑜 ·𝑜 𝑥)))
35		eqeq1 2626	. . . . 5 ⊢ (𝑦 = 𝐴 → (𝑦 = (2𝑜 ·𝑜 𝑥) ↔ 𝐴 = (2𝑜 ·𝑜 𝑥)))
36	35	rexbidv 3052	. . . 4 ⊢ (𝑦 = 𝐴 → (∃𝑥 ∈ ω 𝑦 = (2𝑜 ·𝑜 𝑥) ↔ ∃𝑥 ∈ ω 𝐴 = (2𝑜 ·𝑜 𝑥)))
37	34, 36	imbi12d 334	. . 3 ⊢ (𝑦 = 𝐴 → ((¬ ∃𝑥 ∈ ω suc 𝑦 = (2𝑜 ·𝑜 𝑥) → ∃𝑥 ∈ ω 𝑦 = (2𝑜 ·𝑜 𝑥)) ↔ (¬ ∃𝑥 ∈ ω suc 𝐴 = (2𝑜 ·𝑜 𝑥) → ∃𝑥 ∈ ω 𝐴 = (2𝑜 ·𝑜 𝑥))))
38		peano1 7085	. . . . 5 ⊢ ∅ ∈ ω
39		eqid 2622	. . . . 5 ⊢ ∅ = ∅
40		oveq2 6658	. . . . . . . 8 ⊢ (𝑥 = ∅ → (2𝑜 ·𝑜 𝑥) = (2𝑜 ·𝑜 ∅))
41		om0x 7599	. . . . . . . 8 ⊢ (2𝑜 ·𝑜 ∅) = ∅
42	40, 41	syl6eq 2672	. . . . . . 7 ⊢ (𝑥 = ∅ → (2𝑜 ·𝑜 𝑥) = ∅)
43	42	eqeq2d 2632	. . . . . 6 ⊢ (𝑥 = ∅ → (∅ = (2𝑜 ·𝑜 𝑥) ↔ ∅ = ∅))
44	43	rspcev 3309	. . . . 5 ⊢ ((∅ ∈ ω ∧ ∅ = ∅) → ∃𝑥 ∈ ω ∅ = (2𝑜 ·𝑜 𝑥))
45	38, 39, 44	mp2an 708	. . . 4 ⊢ ∃𝑥 ∈ ω ∅ = (2𝑜 ·𝑜 𝑥)
46	45	a1i 11	. . 3 ⊢ (¬ ∃𝑥 ∈ ω suc ∅ = (2𝑜 ·𝑜 𝑥) → ∃𝑥 ∈ ω ∅ = (2𝑜 ·𝑜 𝑥))
47	1	eqeq2d 2632	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑧 = (2𝑜 ·𝑜 𝑥) ↔ 𝑧 = (2𝑜 ·𝑜 𝑦)))
48	47	cbvrexv 3172	. . . . . 6 ⊢ (∃𝑥 ∈ ω 𝑧 = (2𝑜 ·𝑜 𝑥) ↔ ∃𝑦 ∈ ω 𝑧 = (2𝑜 ·𝑜 𝑦))
49		peano2 7086	. . . . . . . . . 10 ⊢ (𝑦 ∈ ω → suc 𝑦 ∈ ω)
50		2onn 7720	. . . . . . . . . . . 12 ⊢ 2𝑜 ∈ ω
51		nnmsuc 7687	. . . . . . . . . . . 12 ⊢ ((2𝑜 ∈ ω ∧ 𝑦 ∈ ω) → (2𝑜 ·𝑜 suc 𝑦) = ((2𝑜 ·𝑜 𝑦) +𝑜 2𝑜))
52	50, 51	mpan 706	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ω → (2𝑜 ·𝑜 suc 𝑦) = ((2𝑜 ·𝑜 𝑦) +𝑜 2𝑜))
53		df-2o 7561	. . . . . . . . . . . . 13 ⊢ 2𝑜 = suc 1𝑜
54	53	oveq2i 6661	. . . . . . . . . . . 12 ⊢ ((2𝑜 ·𝑜 𝑦) +𝑜 2𝑜) = ((2𝑜 ·𝑜 𝑦) +𝑜 suc 1𝑜)
55		nnmcl 7692	. . . . . . . . . . . . . 14 ⊢ ((2𝑜 ∈ ω ∧ 𝑦 ∈ ω) → (2𝑜 ·𝑜 𝑦) ∈ ω)
56	50, 55	mpan 706	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ω → (2𝑜 ·𝑜 𝑦) ∈ ω)
57		1onn 7719	. . . . . . . . . . . . 13 ⊢ 1𝑜 ∈ ω
58		nnasuc 7686	. . . . . . . . . . . . 13 ⊢ (((2𝑜 ·𝑜 𝑦) ∈ ω ∧ 1𝑜 ∈ ω) → ((2𝑜 ·𝑜 𝑦) +𝑜 suc 1𝑜) = suc ((2𝑜 ·𝑜 𝑦) +𝑜 1𝑜))
59	56, 57, 58	sylancl 694	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ω → ((2𝑜 ·𝑜 𝑦) +𝑜 suc 1𝑜) = suc ((2𝑜 ·𝑜 𝑦) +𝑜 1𝑜))
60	54, 59	syl5req 2669	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ω → suc ((2𝑜 ·𝑜 𝑦) +𝑜 1𝑜) = ((2𝑜 ·𝑜 𝑦) +𝑜 2𝑜))
61		nnon 7071	. . . . . . . . . . . 12 ⊢ ((2𝑜 ·𝑜 𝑦) ∈ ω → (2𝑜 ·𝑜 𝑦) ∈ On)
62		oa1suc 7611	. . . . . . . . . . . 12 ⊢ ((2𝑜 ·𝑜 𝑦) ∈ On → ((2𝑜 ·𝑜 𝑦) +𝑜 1𝑜) = suc (2𝑜 ·𝑜 𝑦))
63		suceq 5790	. . . . . . . . . . . 12 ⊢ (((2𝑜 ·𝑜 𝑦) +𝑜 1𝑜) = suc (2𝑜 ·𝑜 𝑦) → suc ((2𝑜 ·𝑜 𝑦) +𝑜 1𝑜) = suc suc (2𝑜 ·𝑜 𝑦))
64	56, 61, 62, 63	4syl 19	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ω → suc ((2𝑜 ·𝑜 𝑦) +𝑜 1𝑜) = suc suc (2𝑜 ·𝑜 𝑦))
65	52, 60, 64	3eqtr2rd 2663	. . . . . . . . . 10 ⊢ (𝑦 ∈ ω → suc suc (2𝑜 ·𝑜 𝑦) = (2𝑜 ·𝑜 suc 𝑦))
66		oveq2 6658	. . . . . . . . . . . 12 ⊢ (𝑥 = suc 𝑦 → (2𝑜 ·𝑜 𝑥) = (2𝑜 ·𝑜 suc 𝑦))
67	66	eqeq2d 2632	. . . . . . . . . . 11 ⊢ (𝑥 = suc 𝑦 → (suc suc (2𝑜 ·𝑜 𝑦) = (2𝑜 ·𝑜 𝑥) ↔ suc suc (2𝑜 ·𝑜 𝑦) = (2𝑜 ·𝑜 suc 𝑦)))
68	67	rspcev 3309	. . . . . . . . . 10 ⊢ ((suc 𝑦 ∈ ω ∧ suc suc (2𝑜 ·𝑜 𝑦) = (2𝑜 ·𝑜 suc 𝑦)) → ∃𝑥 ∈ ω suc suc (2𝑜 ·𝑜 𝑦) = (2𝑜 ·𝑜 𝑥))
69	49, 65, 68	syl2anc 693	. . . . . . . . 9 ⊢ (𝑦 ∈ ω → ∃𝑥 ∈ ω suc suc (2𝑜 ·𝑜 𝑦) = (2𝑜 ·𝑜 𝑥))
70		suceq 5790	. . . . . . . . . . . 12 ⊢ (𝑧 = (2𝑜 ·𝑜 𝑦) → suc 𝑧 = suc (2𝑜 ·𝑜 𝑦))
71		suceq 5790	. . . . . . . . . . . 12 ⊢ (suc 𝑧 = suc (2𝑜 ·𝑜 𝑦) → suc suc 𝑧 = suc suc (2𝑜 ·𝑜 𝑦))
72	70, 71	syl 17	. . . . . . . . . . 11 ⊢ (𝑧 = (2𝑜 ·𝑜 𝑦) → suc suc 𝑧 = suc suc (2𝑜 ·𝑜 𝑦))
73	72	eqeq1d 2624	. . . . . . . . . 10 ⊢ (𝑧 = (2𝑜 ·𝑜 𝑦) → (suc suc 𝑧 = (2𝑜 ·𝑜 𝑥) ↔ suc suc (2𝑜 ·𝑜 𝑦) = (2𝑜 ·𝑜 𝑥)))
74	73	rexbidv 3052	. . . . . . . . 9 ⊢ (𝑧 = (2𝑜 ·𝑜 𝑦) → (∃𝑥 ∈ ω suc suc 𝑧 = (2𝑜 ·𝑜 𝑥) ↔ ∃𝑥 ∈ ω suc suc (2𝑜 ·𝑜 𝑦) = (2𝑜 ·𝑜 𝑥)))
75	69, 74	syl5ibrcom 237	. . . . . . . 8 ⊢ (𝑦 ∈ ω → (𝑧 = (2𝑜 ·𝑜 𝑦) → ∃𝑥 ∈ ω suc suc 𝑧 = (2𝑜 ·𝑜 𝑥)))
76	75	rexlimiv 3027	. . . . . . 7 ⊢ (∃𝑦 ∈ ω 𝑧 = (2𝑜 ·𝑜 𝑦) → ∃𝑥 ∈ ω suc suc 𝑧 = (2𝑜 ·𝑜 𝑥))
77	76	a1i 11	. . . . . 6 ⊢ (𝑧 ∈ ω → (∃𝑦 ∈ ω 𝑧 = (2𝑜 ·𝑜 𝑦) → ∃𝑥 ∈ ω suc suc 𝑧 = (2𝑜 ·𝑜 𝑥)))
78	48, 77	syl5bi 232	. . . . 5 ⊢ (𝑧 ∈ ω → (∃𝑥 ∈ ω 𝑧 = (2𝑜 ·𝑜 𝑥) → ∃𝑥 ∈ ω suc suc 𝑧 = (2𝑜 ·𝑜 𝑥)))
79	78	con3d 148	. . . 4 ⊢ (𝑧 ∈ ω → (¬ ∃𝑥 ∈ ω suc suc 𝑧 = (2𝑜 ·𝑜 𝑥) → ¬ ∃𝑥 ∈ ω 𝑧 = (2𝑜 ·𝑜 𝑥)))
80		con1 143	. . . 4 ⊢ ((¬ ∃𝑥 ∈ ω suc 𝑧 = (2𝑜 ·𝑜 𝑥) → ∃𝑥 ∈ ω 𝑧 = (2𝑜 ·𝑜 𝑥)) → (¬ ∃𝑥 ∈ ω 𝑧 = (2𝑜 ·𝑜 𝑥) → ∃𝑥 ∈ ω suc 𝑧 = (2𝑜 ·𝑜 𝑥)))
81	79, 80	syl9 77	. . 3 ⊢ (𝑧 ∈ ω → ((¬ ∃𝑥 ∈ ω suc 𝑧 = (2𝑜 ·𝑜 𝑥) → ∃𝑥 ∈ ω 𝑧 = (2𝑜 ·𝑜 𝑥)) → (¬ ∃𝑥 ∈ ω suc suc 𝑧 = (2𝑜 ·𝑜 𝑥) → ∃𝑥 ∈ ω suc 𝑧 = (2𝑜 ·𝑜 𝑥))))
82	16, 23, 30, 37, 46, 81	finds 7092	. 2 ⊢ (𝐴 ∈ ω → (¬ ∃𝑥 ∈ ω suc 𝐴 = (2𝑜 ·𝑜 𝑥) → ∃𝑥 ∈ ω 𝐴 = (2𝑜 ·𝑜 𝑥)))
83	9, 82	impbid2 216	1 ⊢ (𝐴 ∈ ω → (∃𝑥 ∈ ω 𝐴 = (2𝑜 ·𝑜 𝑥) ↔ ¬ ∃𝑥 ∈ ω suc 𝐴 = (2𝑜 ·𝑜 𝑥)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∃wrex 2913  ∅c0 3915  Oncon0 5723  suc csuc 5725  (class class class)co 6650  ωcom 7065  1_𝑜c1o 7553  2_𝑜c2o 7554   +_𝑜 coa 7557   ·_𝑜 comu 7558

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-omul 7565

This theorem is referenced by:  fin1a2lem5  9226