Theorem nneob 8673

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 7418	. . . . 5 ⊢ (𝑥 = 𝑦 → (2o ·o 𝑥) = (2o ·o 𝑦))
2	1	eqeq2d 2747	. . . 4 ⊢ (𝑥 = 𝑦 → (𝐴 = (2o ·o 𝑥) ↔ 𝐴 = (2o ·o 𝑦)))
3	2	cbvrexvw 3225	. . 3 ⊢ (∃𝑥 ∈ ω 𝐴 = (2o ·o 𝑥) ↔ ∃𝑦 ∈ ω 𝐴 = (2o ·o 𝑦))
4		nnneo 8672	. . . . . . 7 ⊢ ((𝑦 ∈ ω ∧ 𝑥 ∈ ω ∧ 𝐴 = (2o ·o 𝑦)) → ¬ suc 𝐴 = (2o ·o 𝑥))
5	4	3com23 1126	. . . . . 6 ⊢ ((𝑦 ∈ ω ∧ 𝐴 = (2o ·o 𝑦) ∧ 𝑥 ∈ ω) → ¬ suc 𝐴 = (2o ·o 𝑥))
6	5	3expa 1118	. . . . 5 ⊢ (((𝑦 ∈ ω ∧ 𝐴 = (2o ·o 𝑦)) ∧ 𝑥 ∈ ω) → ¬ suc 𝐴 = (2o ·o 𝑥))
7	6	nrexdv 3136	. . . 4 ⊢ ((𝑦 ∈ ω ∧ 𝐴 = (2o ·o 𝑦)) → ¬ ∃𝑥 ∈ ω suc 𝐴 = (2o ·o 𝑥))
8	7	rexlimiva 3134	. . 3 ⊢ (∃𝑦 ∈ ω 𝐴 = (2o ·o 𝑦) → ¬ ∃𝑥 ∈ ω suc 𝐴 = (2o ·o 𝑥))
9	3, 8	sylbi 217	. 2 ⊢ (∃𝑥 ∈ ω 𝐴 = (2o ·o 𝑥) → ¬ ∃𝑥 ∈ ω suc 𝐴 = (2o ·o 𝑥))
10		suceq 6424	. . . . . . 7 ⊢ (𝑦 = ∅ → suc 𝑦 = suc ∅)
11	10	eqeq1d 2738	. . . . . 6 ⊢ (𝑦 = ∅ → (suc 𝑦 = (2o ·o 𝑥) ↔ suc ∅ = (2o ·o 𝑥)))
12	11	rexbidv 3165	. . . . 5 ⊢ (𝑦 = ∅ → (∃𝑥 ∈ ω suc 𝑦 = (2o ·o 𝑥) ↔ ∃𝑥 ∈ ω suc ∅ = (2o ·o 𝑥)))
13	12	notbid 318	. . . 4 ⊢ (𝑦 = ∅ → (¬ ∃𝑥 ∈ ω suc 𝑦 = (2o ·o 𝑥) ↔ ¬ ∃𝑥 ∈ ω suc ∅ = (2o ·o 𝑥)))
14		eqeq1 2740	. . . . 5 ⊢ (𝑦 = ∅ → (𝑦 = (2o ·o 𝑥) ↔ ∅ = (2o ·o 𝑥)))
15	14	rexbidv 3165	. . . 4 ⊢ (𝑦 = ∅ → (∃𝑥 ∈ ω 𝑦 = (2o ·o 𝑥) ↔ ∃𝑥 ∈ ω ∅ = (2o ·o 𝑥)))
16	13, 15	imbi12d 344	. . 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 6424	. . . . . . 7 ⊢ (𝑦 = 𝑧 → suc 𝑦 = suc 𝑧)
18	17	eqeq1d 2738	. . . . . 6 ⊢ (𝑦 = 𝑧 → (suc 𝑦 = (2o ·o 𝑥) ↔ suc 𝑧 = (2o ·o 𝑥)))
19	18	rexbidv 3165	. . . . 5 ⊢ (𝑦 = 𝑧 → (∃𝑥 ∈ ω suc 𝑦 = (2o ·o 𝑥) ↔ ∃𝑥 ∈ ω suc 𝑧 = (2o ·o 𝑥)))
20	19	notbid 318	. . . 4 ⊢ (𝑦 = 𝑧 → (¬ ∃𝑥 ∈ ω suc 𝑦 = (2o ·o 𝑥) ↔ ¬ ∃𝑥 ∈ ω suc 𝑧 = (2o ·o 𝑥)))
21		eqeq1 2740	. . . . 5 ⊢ (𝑦 = 𝑧 → (𝑦 = (2o ·o 𝑥) ↔ 𝑧 = (2o ·o 𝑥)))
22	21	rexbidv 3165	. . . 4 ⊢ (𝑦 = 𝑧 → (∃𝑥 ∈ ω 𝑦 = (2o ·o 𝑥) ↔ ∃𝑥 ∈ ω 𝑧 = (2o ·o 𝑥)))
23	20, 22	imbi12d 344	. . 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 6424	. . . . . . 7 ⊢ (𝑦 = suc 𝑧 → suc 𝑦 = suc suc 𝑧)
25	24	eqeq1d 2738	. . . . . 6 ⊢ (𝑦 = suc 𝑧 → (suc 𝑦 = (2o ·o 𝑥) ↔ suc suc 𝑧 = (2o ·o 𝑥)))
26	25	rexbidv 3165	. . . . 5 ⊢ (𝑦 = suc 𝑧 → (∃𝑥 ∈ ω suc 𝑦 = (2o ·o 𝑥) ↔ ∃𝑥 ∈ ω suc suc 𝑧 = (2o ·o 𝑥)))
27	26	notbid 318	. . . 4 ⊢ (𝑦 = suc 𝑧 → (¬ ∃𝑥 ∈ ω suc 𝑦 = (2o ·o 𝑥) ↔ ¬ ∃𝑥 ∈ ω suc suc 𝑧 = (2o ·o 𝑥)))
28		eqeq1 2740	. . . . 5 ⊢ (𝑦 = suc 𝑧 → (𝑦 = (2o ·o 𝑥) ↔ suc 𝑧 = (2o ·o 𝑥)))
29	28	rexbidv 3165	. . . 4 ⊢ (𝑦 = suc 𝑧 → (∃𝑥 ∈ ω 𝑦 = (2o ·o 𝑥) ↔ ∃𝑥 ∈ ω suc 𝑧 = (2o ·o 𝑥)))
30	27, 29	imbi12d 344	. . 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 6424	. . . . . . 7 ⊢ (𝑦 = 𝐴 → suc 𝑦 = suc 𝐴)
32	31	eqeq1d 2738	. . . . . 6 ⊢ (𝑦 = 𝐴 → (suc 𝑦 = (2o ·o 𝑥) ↔ suc 𝐴 = (2o ·o 𝑥)))
33	32	rexbidv 3165	. . . . 5 ⊢ (𝑦 = 𝐴 → (∃𝑥 ∈ ω suc 𝑦 = (2o ·o 𝑥) ↔ ∃𝑥 ∈ ω suc 𝐴 = (2o ·o 𝑥)))
34	33	notbid 318	. . . 4 ⊢ (𝑦 = 𝐴 → (¬ ∃𝑥 ∈ ω suc 𝑦 = (2o ·o 𝑥) ↔ ¬ ∃𝑥 ∈ ω suc 𝐴 = (2o ·o 𝑥)))
35		eqeq1 2740	. . . . 5 ⊢ (𝑦 = 𝐴 → (𝑦 = (2o ·o 𝑥) ↔ 𝐴 = (2o ·o 𝑥)))
36	35	rexbidv 3165	. . . 4 ⊢ (𝑦 = 𝐴 → (∃𝑥 ∈ ω 𝑦 = (2o ·o 𝑥) ↔ ∃𝑥 ∈ ω 𝐴 = (2o ·o 𝑥)))
37	34, 36	imbi12d 344	. . 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 7889	. . . . 5 ⊢ ∅ ∈ ω
39		eqid 2736	. . . . 5 ⊢ ∅ = ∅
40		oveq2 7418	. . . . . . 7 ⊢ (𝑥 = ∅ → (2o ·o 𝑥) = (2o ·o ∅))
41		2on 8499	. . . . . . . 8 ⊢ 2o ∈ On
42		om0 8534	. . . . . . . 8 ⊢ (2o ∈ On → (2o ·o ∅) = ∅)
43	41, 42	ax-mp 5	. . . . . . 7 ⊢ (2o ·o ∅) = ∅
44	40, 43	eqtrdi 2787	. . . . . 6 ⊢ (𝑥 = ∅ → (2o ·o 𝑥) = ∅)
45	44	rspceeqv 3629	. . . . 5 ⊢ ((∅ ∈ ω ∧ ∅ = ∅) → ∃𝑥 ∈ ω ∅ = (2o ·o 𝑥))
46	38, 39, 45	mp2an 692	. . . 4 ⊢ ∃𝑥 ∈ ω ∅ = (2o ·o 𝑥)
47	46	a1i 11	. . 3 ⊢ (¬ ∃𝑥 ∈ ω suc ∅ = (2o ·o 𝑥) → ∃𝑥 ∈ ω ∅ = (2o ·o 𝑥))
48	1	eqeq2d 2747	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑧 = (2o ·o 𝑥) ↔ 𝑧 = (2o ·o 𝑦)))
49	48	cbvrexvw 3225	. . . . . 6 ⊢ (∃𝑥 ∈ ω 𝑧 = (2o ·o 𝑥) ↔ ∃𝑦 ∈ ω 𝑧 = (2o ·o 𝑦))
50		peano2 7891	. . . . . . . . . 10 ⊢ (𝑦 ∈ ω → suc 𝑦 ∈ ω)
51		2onn 8659	. . . . . . . . . . . 12 ⊢ 2o ∈ ω
52		nnmsuc 8624	. . . . . . . . . . . 12 ⊢ ((2o ∈ ω ∧ 𝑦 ∈ ω) → (2o ·o suc 𝑦) = ((2o ·o 𝑦) +o 2o))
53	51, 52	mpan 690	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ω → (2o ·o suc 𝑦) = ((2o ·o 𝑦) +o 2o))
54		df-2o 8486	. . . . . . . . . . . . 13 ⊢ 2o = suc 1o
55	54	oveq2i 7421	. . . . . . . . . . . 12 ⊢ ((2o ·o 𝑦) +o 2o) = ((2o ·o 𝑦) +o suc 1o)
56		nnmcl 8629	. . . . . . . . . . . . . 14 ⊢ ((2o ∈ ω ∧ 𝑦 ∈ ω) → (2o ·o 𝑦) ∈ ω)
57	51, 56	mpan 690	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ω → (2o ·o 𝑦) ∈ ω)
58		1onn 8657	. . . . . . . . . . . . 13 ⊢ 1o ∈ ω
59		nnasuc 8623	. . . . . . . . . . . . 13 ⊢ (((2o ·o 𝑦) ∈ ω ∧ 1o ∈ ω) → ((2o ·o 𝑦) +o suc 1o) = suc ((2o ·o 𝑦) +o 1o))
60	57, 58, 59	sylancl 586	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ω → ((2o ·o 𝑦) +o suc 1o) = suc ((2o ·o 𝑦) +o 1o))
61	55, 60	eqtr2id 2784	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ω → suc ((2o ·o 𝑦) +o 1o) = ((2o ·o 𝑦) +o 2o))
62		nnon 7872	. . . . . . . . . . . 12 ⊢ ((2o ·o 𝑦) ∈ ω → (2o ·o 𝑦) ∈ On)
63		oa1suc 8548	. . . . . . . . . . . 12 ⊢ ((2o ·o 𝑦) ∈ On → ((2o ·o 𝑦) +o 1o) = suc (2o ·o 𝑦))
64		suceq 6424	. . . . . . . . . . . 12 ⊢ (((2o ·o 𝑦) +o 1o) = suc (2o ·o 𝑦) → suc ((2o ·o 𝑦) +o 1o) = suc suc (2o ·o 𝑦))
65	57, 62, 63, 64	4syl 19	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ω → suc ((2o ·o 𝑦) +o 1o) = suc suc (2o ·o 𝑦))
66	53, 61, 65	3eqtr2rd 2778	. . . . . . . . . 10 ⊢ (𝑦 ∈ ω → suc suc (2o ·o 𝑦) = (2o ·o suc 𝑦))
67		oveq2 7418	. . . . . . . . . . 11 ⊢ (𝑥 = suc 𝑦 → (2o ·o 𝑥) = (2o ·o suc 𝑦))
68	67	rspceeqv 3629	. . . . . . . . . 10 ⊢ ((suc 𝑦 ∈ ω ∧ suc suc (2o ·o 𝑦) = (2o ·o suc 𝑦)) → ∃𝑥 ∈ ω suc suc (2o ·o 𝑦) = (2o ·o 𝑥))
69	50, 66, 68	syl2anc 584	. . . . . . . . 9 ⊢ (𝑦 ∈ ω → ∃𝑥 ∈ ω suc suc (2o ·o 𝑦) = (2o ·o 𝑥))
70		suceq 6424	. . . . . . . . . . . 12 ⊢ (𝑧 = (2o ·o 𝑦) → suc 𝑧 = suc (2o ·o 𝑦))
71		suceq 6424	. . . . . . . . . . . 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 2738	. . . . . . . . . 10 ⊢ (𝑧 = (2o ·o 𝑦) → (suc suc 𝑧 = (2o ·o 𝑥) ↔ suc suc (2o ·o 𝑦) = (2o ·o 𝑥)))
74	73	rexbidv 3165	. . . . . . . . 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 247	. . . . . . . 8 ⊢ (𝑦 ∈ ω → (𝑧 = (2o ·o 𝑦) → ∃𝑥 ∈ ω suc suc 𝑧 = (2o ·o 𝑥)))
76	75	rexlimiv 3135	. . . . . . 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	49, 77	biimtrid 242	. . . . 5 ⊢ (𝑧 ∈ ω → (∃𝑥 ∈ ω 𝑧 = (2o ·o 𝑥) → ∃𝑥 ∈ ω suc suc 𝑧 = (2o ·o 𝑥)))
79	78	con3d 152	. . . 4 ⊢ (𝑧 ∈ ω → (¬ ∃𝑥 ∈ ω suc suc 𝑧 = (2o ·o 𝑥) → ¬ ∃𝑥 ∈ ω 𝑧 = (2o ·o 𝑥)))
80		con1 146	. . . 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, 47, 81	finds 7897	. 2 ⊢ (𝐴 ∈ ω → (¬ ∃𝑥 ∈ ω suc 𝐴 = (2o ·o 𝑥) → ∃𝑥 ∈ ω 𝐴 = (2o ·o 𝑥)))
83	9, 82	impbid2 226	1 ⊢ (𝐴 ∈ ω → (∃𝑥 ∈ ω 𝐴 = (2o ·o 𝑥) ↔ ¬ ∃𝑥 ∈ ω suc 𝐴 = (2o ·o 𝑥)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∃wrex 3061  ∅c0 4313  Oncon0 6357  suc csuc 6359  (class class class)co 7410  ωcom 7866  1_oc1o 8478  2_oc2o 8479   +_o coa 8482   ·_o comu 8483

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pr 5407  ax-un 7734

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-tr 5235  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6295  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7867  df-2nd 7994  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-1o 8485  df-2o 8486  df-oadd 8489  df-omul 8490

This theorem is referenced by:  fin1a2lem5  10423