Theorem nnneo 8693

Description: If a natural number is even, its successor is odd. (Contributed by Mario Carneiro, 16-Nov-2014.)

Assertion

Ref	Expression
nnneo	⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 = (2o ·o 𝐴)) → ¬ suc 𝐶 = (2o ·o 𝐵))

Proof of Theorem nnneo

Step	Hyp	Ref	Expression
1		nnon 7893	. . . 4 ⊢ (𝐴 ∈ ω → 𝐴 ∈ On)
2		onnbtwn 6478	. . . 4 ⊢ (𝐴 ∈ On → ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ suc 𝐴))
3	1, 2	syl 17	. . 3 ⊢ (𝐴 ∈ ω → ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ suc 𝐴))
4	3	3ad2ant1 1134	. 2 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 = (2o ·o 𝐴)) → ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ suc 𝐴))
5		suceq 6450	. . . . 5 ⊢ (𝐶 = (2o ·o 𝐴) → suc 𝐶 = suc (2o ·o 𝐴))
6	5	eqeq1d 2739	. . . 4 ⊢ (𝐶 = (2o ·o 𝐴) → (suc 𝐶 = (2o ·o 𝐵) ↔ suc (2o ·o 𝐴) = (2o ·o 𝐵)))
7	6	3ad2ant3 1136	. . 3 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 = (2o ·o 𝐴)) → (suc 𝐶 = (2o ·o 𝐵) ↔ suc (2o ·o 𝐴) = (2o ·o 𝐵)))
8		ovex 7464	. . . . . . . 8 ⊢ (2o ·o 𝐴) ∈ V
9	8	sucid 6466	. . . . . . 7 ⊢ (2o ·o 𝐴) ∈ suc (2o ·o 𝐴)
10		eleq2 2830	. . . . . . 7 ⊢ (suc (2o ·o 𝐴) = (2o ·o 𝐵) → ((2o ·o 𝐴) ∈ suc (2o ·o 𝐴) ↔ (2o ·o 𝐴) ∈ (2o ·o 𝐵)))
11	9, 10	mpbii 233	. . . . . 6 ⊢ (suc (2o ·o 𝐴) = (2o ·o 𝐵) → (2o ·o 𝐴) ∈ (2o ·o 𝐵))
12		2onn 8680	. . . . . . . 8 ⊢ 2o ∈ ω
13		nnmord 8670	. . . . . . . 8 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 2o ∈ ω) → ((𝐴 ∈ 𝐵 ∧ ∅ ∈ 2o) ↔ (2o ·o 𝐴) ∈ (2o ·o 𝐵)))
14	12, 13	mp3an3 1452	. . . . . . 7 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐴 ∈ 𝐵 ∧ ∅ ∈ 2o) ↔ (2o ·o 𝐴) ∈ (2o ·o 𝐵)))
15		simpl 482	. . . . . . 7 ⊢ ((𝐴 ∈ 𝐵 ∧ ∅ ∈ 2o) → 𝐴 ∈ 𝐵)
16	14, 15	biimtrrdi 254	. . . . . 6 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((2o ·o 𝐴) ∈ (2o ·o 𝐵) → 𝐴 ∈ 𝐵))
17	11, 16	syl5 34	. . . . 5 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (suc (2o ·o 𝐴) = (2o ·o 𝐵) → 𝐴 ∈ 𝐵))
18		simpr 484	. . . . . . . . 9 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ suc (2o ·o 𝐴) = (2o ·o 𝐵)) → suc (2o ·o 𝐴) = (2o ·o 𝐵))
19		nnmcl 8650	. . . . . . . . . . . . 13 ⊢ ((2o ∈ ω ∧ 𝐴 ∈ ω) → (2o ·o 𝐴) ∈ ω)
20	12, 19	mpan 690	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ω → (2o ·o 𝐴) ∈ ω)
21		nnon 7893	. . . . . . . . . . . 12 ⊢ ((2o ·o 𝐴) ∈ ω → (2o ·o 𝐴) ∈ On)
22		oa1suc 8569	. . . . . . . . . . . 12 ⊢ ((2o ·o 𝐴) ∈ On → ((2o ·o 𝐴) +o 1o) = suc (2o ·o 𝐴))
23	20, 21, 22	3syl 18	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ω → ((2o ·o 𝐴) +o 1o) = suc (2o ·o 𝐴))
24		1oex 8516	. . . . . . . . . . . . . . 15 ⊢ 1o ∈ V
25	24	sucid 6466	. . . . . . . . . . . . . 14 ⊢ 1o ∈ suc 1o
26		df-2o 8507	. . . . . . . . . . . . . 14 ⊢ 2o = suc 1o
27	25, 26	eleqtrri 2840	. . . . . . . . . . . . 13 ⊢ 1o ∈ 2o
28		1onn 8678	. . . . . . . . . . . . . 14 ⊢ 1o ∈ ω
29		nnaord 8657	. . . . . . . . . . . . . 14 ⊢ ((1o ∈ ω ∧ 2o ∈ ω ∧ (2o ·o 𝐴) ∈ ω) → (1o ∈ 2o ↔ ((2o ·o 𝐴) +o 1o) ∈ ((2o ·o 𝐴) +o 2o)))
30	28, 12, 20, 29	mp3an12i 1467	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ω → (1o ∈ 2o ↔ ((2o ·o 𝐴) +o 1o) ∈ ((2o ·o 𝐴) +o 2o)))
31	27, 30	mpbii 233	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ω → ((2o ·o 𝐴) +o 1o) ∈ ((2o ·o 𝐴) +o 2o))
32		nnmsuc 8645	. . . . . . . . . . . . 13 ⊢ ((2o ∈ ω ∧ 𝐴 ∈ ω) → (2o ·o suc 𝐴) = ((2o ·o 𝐴) +o 2o))
33	12, 32	mpan 690	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ω → (2o ·o suc 𝐴) = ((2o ·o 𝐴) +o 2o))
34	31, 33	eleqtrrd 2844	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ω → ((2o ·o 𝐴) +o 1o) ∈ (2o ·o suc 𝐴))
35	23, 34	eqeltrrd 2842	. . . . . . . . . 10 ⊢ (𝐴 ∈ ω → suc (2o ·o 𝐴) ∈ (2o ·o suc 𝐴))
36	35	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ suc (2o ·o 𝐴) = (2o ·o 𝐵)) → suc (2o ·o 𝐴) ∈ (2o ·o suc 𝐴))
37	18, 36	eqeltrrd 2842	. . . . . . . 8 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ suc (2o ·o 𝐴) = (2o ·o 𝐵)) → (2o ·o 𝐵) ∈ (2o ·o suc 𝐴))
38		peano2 7912	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ω → suc 𝐴 ∈ ω)
39		nnmord 8670	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ ω ∧ suc 𝐴 ∈ ω ∧ 2o ∈ ω) → ((𝐵 ∈ suc 𝐴 ∧ ∅ ∈ 2o) ↔ (2o ·o 𝐵) ∈ (2o ·o suc 𝐴)))
40	12, 39	mp3an3 1452	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ ω ∧ suc 𝐴 ∈ ω) → ((𝐵 ∈ suc 𝐴 ∧ ∅ ∈ 2o) ↔ (2o ·o 𝐵) ∈ (2o ·o suc 𝐴)))
41	38, 40	sylan2 593	. . . . . . . . . 10 ⊢ ((𝐵 ∈ ω ∧ 𝐴 ∈ ω) → ((𝐵 ∈ suc 𝐴 ∧ ∅ ∈ 2o) ↔ (2o ·o 𝐵) ∈ (2o ·o suc 𝐴)))
42	41	ancoms 458	. . . . . . . . 9 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐵 ∈ suc 𝐴 ∧ ∅ ∈ 2o) ↔ (2o ·o 𝐵) ∈ (2o ·o suc 𝐴)))
43	42	adantr 480	. . . . . . . 8 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ suc (2o ·o 𝐴) = (2o ·o 𝐵)) → ((𝐵 ∈ suc 𝐴 ∧ ∅ ∈ 2o) ↔ (2o ·o 𝐵) ∈ (2o ·o suc 𝐴)))
44	37, 43	mpbird 257	. . . . . . 7 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ suc (2o ·o 𝐴) = (2o ·o 𝐵)) → (𝐵 ∈ suc 𝐴 ∧ ∅ ∈ 2o))
45	44	simpld 494	. . . . . 6 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ suc (2o ·o 𝐴) = (2o ·o 𝐵)) → 𝐵 ∈ suc 𝐴)
46	45	ex 412	. . . . 5 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (suc (2o ·o 𝐴) = (2o ·o 𝐵) → 𝐵 ∈ suc 𝐴))
47	17, 46	jcad 512	. . . 4 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (suc (2o ·o 𝐴) = (2o ·o 𝐵) → (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ suc 𝐴)))
48	47	3adant3 1133	. . 3 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 = (2o ·o 𝐴)) → (suc (2o ·o 𝐴) = (2o ·o 𝐵) → (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ suc 𝐴)))
49	7, 48	sylbid 240	. 2 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 = (2o ·o 𝐴)) → (suc 𝐶 = (2o ·o 𝐵) → (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ suc 𝐴)))
50	4, 49	mtod 198	1 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 = (2o ·o 𝐴)) → ¬ suc 𝐶 = (2o ·o 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1540   ∈ wcel 2108  ∅c0 4333  Oncon0 6384  suc csuc 6386  (class class class)co 7431  ωcom 7887  1_oc1o 8499  2_oc2o 8500   +_o coa 8503   ·_o comu 8504

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-oadd 8510  df-omul 8511

This theorem is referenced by:  nneob  8694