Theorem nnneo 7618

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 6963	. . . 4 ⊢ (𝐴 ∈ ω → 𝐴 ∈ On)
2		onnbtwn 5735	. . . 4 ⊢ (𝐴 ∈ On → ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ suc 𝐴))
3	1, 2	syl 17	. . 3 ⊢ (𝐴 ∈ ω → ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ suc 𝐴))
4	3	3ad2ant1 1075	. 2 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 = (2𝑜 ·𝑜 𝐴)) → ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ suc 𝐴))
5		suceq 5707	. . . . 5 ⊢ (𝐶 = (2𝑜 ·𝑜 𝐴) → suc 𝐶 = suc (2𝑜 ·𝑜 𝐴))
6	5	eqeq1d 2612	. . . 4 ⊢ (𝐶 = (2𝑜 ·𝑜 𝐴) → (suc 𝐶 = (2𝑜 ·𝑜 𝐵) ↔ suc (2𝑜 ·𝑜 𝐴) = (2𝑜 ·𝑜 𝐵)))
7	6	3ad2ant3 1077	. . 3 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 = (2𝑜 ·𝑜 𝐴)) → (suc 𝐶 = (2𝑜 ·𝑜 𝐵) ↔ suc (2𝑜 ·𝑜 𝐴) = (2𝑜 ·𝑜 𝐵)))
8		ovex 6577	. . . . . . . 8 ⊢ (2𝑜 ·𝑜 𝐴) ∈ V
9	8	sucid 5721	. . . . . . 7 ⊢ (2𝑜 ·𝑜 𝐴) ∈ suc (2𝑜 ·𝑜 𝐴)
10		eleq2 2677	. . . . . . 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 222	. . . . . 6 ⊢ (suc (2𝑜 ·𝑜 𝐴) = (2𝑜 ·𝑜 𝐵) → (2𝑜 ·𝑜 𝐴) ∈ (2𝑜 ·𝑜 𝐵))
12		2onn 7607	. . . . . . . 8 ⊢ 2𝑜 ∈ ω
13		nnmord 7599	. . . . . . . 8 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 2𝑜 ∈ ω) → ((𝐴 ∈ 𝐵 ∧ ∅ ∈ 2𝑜) ↔ (2𝑜 ·𝑜 𝐴) ∈ (2𝑜 ·𝑜 𝐵)))
14	12, 13	mp3an3 1405	. . . . . . 7 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐴 ∈ 𝐵 ∧ ∅ ∈ 2𝑜) ↔ (2𝑜 ·𝑜 𝐴) ∈ (2𝑜 ·𝑜 𝐵)))
15		simpl 472	. . . . . . 7 ⊢ ((𝐴 ∈ 𝐵 ∧ ∅ ∈ 2𝑜) → 𝐴 ∈ 𝐵)
16	14, 15	syl6bir 243	. . . . . 6 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((2𝑜 ·𝑜 𝐴) ∈ (2𝑜 ·𝑜 𝐵) → 𝐴 ∈ 𝐵))
17	11, 16	syl5 33	. . . . 5 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (suc (2𝑜 ·𝑜 𝐴) = (2𝑜 ·𝑜 𝐵) → 𝐴 ∈ 𝐵))
18		simpr 476	. . . . . . . . 9 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ suc (2𝑜 ·𝑜 𝐴) = (2𝑜 ·𝑜 𝐵)) → suc (2𝑜 ·𝑜 𝐴) = (2𝑜 ·𝑜 𝐵))
19		nnmcl 7579	. . . . . . . . . . . . 13 ⊢ ((2𝑜 ∈ ω ∧ 𝐴 ∈ ω) → (2𝑜 ·𝑜 𝐴) ∈ ω)
20	12, 19	mpan 702	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ω → (2𝑜 ·𝑜 𝐴) ∈ ω)
21		nnon 6963	. . . . . . . . . . . 12 ⊢ ((2𝑜 ·𝑜 𝐴) ∈ ω → (2𝑜 ·𝑜 𝐴) ∈ On)
22		oa1suc 7498	. . . . . . . . . . . 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		1onn 7606	. . . . . . . . . . . . . . . 16 ⊢ 1𝑜 ∈ ω
25	24	elexi 3186	. . . . . . . . . . . . . . 15 ⊢ 1𝑜 ∈ V
26	25	sucid 5721	. . . . . . . . . . . . . 14 ⊢ 1𝑜 ∈ suc 1𝑜
27		df-2o 7448	. . . . . . . . . . . . . 14 ⊢ 2𝑜 = suc 1𝑜
28	26, 27	eleqtrri 2687	. . . . . . . . . . . . 13 ⊢ 1𝑜 ∈ 2𝑜
29		nnaord 7586	. . . . . . . . . . . . . . 15 ⊢ ((1𝑜 ∈ ω ∧ 2𝑜 ∈ ω ∧ (2𝑜 ·𝑜 𝐴) ∈ ω) → (1𝑜 ∈ 2𝑜 ↔ ((2𝑜 ·𝑜 𝐴) +𝑜 1𝑜) ∈ ((2𝑜 ·𝑜 𝐴) +𝑜 2𝑜)))
30	24, 12, 29	mp3an12 1406	. . . . . . . . . . . . . 14 ⊢ ((2𝑜 ·𝑜 𝐴) ∈ ω → (1𝑜 ∈ 2𝑜 ↔ ((2𝑜 ·𝑜 𝐴) +𝑜 1𝑜) ∈ ((2𝑜 ·𝑜 𝐴) +𝑜 2𝑜)))
31	20, 30	syl 17	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ω → (1𝑜 ∈ 2𝑜 ↔ ((2𝑜 ·𝑜 𝐴) +𝑜 1𝑜) ∈ ((2𝑜 ·𝑜 𝐴) +𝑜 2𝑜)))
32	28, 31	mpbii 222	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ω → ((2𝑜 ·𝑜 𝐴) +𝑜 1𝑜) ∈ ((2𝑜 ·𝑜 𝐴) +𝑜 2𝑜))
33		nnmsuc 7574	. . . . . . . . . . . . 13 ⊢ ((2𝑜 ∈ ω ∧ 𝐴 ∈ ω) → (2𝑜 ·𝑜 suc 𝐴) = ((2𝑜 ·𝑜 𝐴) +𝑜 2𝑜))
34	12, 33	mpan 702	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ω → (2𝑜 ·𝑜 suc 𝐴) = ((2𝑜 ·𝑜 𝐴) +𝑜 2𝑜))
35	32, 34	eleqtrrd 2691	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ω → ((2𝑜 ·𝑜 𝐴) +𝑜 1𝑜) ∈ (2𝑜 ·𝑜 suc 𝐴))
36	23, 35	eqeltrrd 2689	. . . . . . . . . 10 ⊢ (𝐴 ∈ ω → suc (2𝑜 ·𝑜 𝐴) ∈ (2𝑜 ·𝑜 suc 𝐴))
37	36	ad2antrr 758	. . . . . . . . 9 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ suc (2𝑜 ·𝑜 𝐴) = (2𝑜 ·𝑜 𝐵)) → suc (2𝑜 ·𝑜 𝐴) ∈ (2𝑜 ·𝑜 suc 𝐴))
38	18, 37	eqeltrrd 2689	. . . . . . . 8 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ suc (2𝑜 ·𝑜 𝐴) = (2𝑜 ·𝑜 𝐵)) → (2𝑜 ·𝑜 𝐵) ∈ (2𝑜 ·𝑜 suc 𝐴))
39		peano2 6978	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ω → suc 𝐴 ∈ ω)
40		nnmord 7599	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ ω ∧ suc 𝐴 ∈ ω ∧ 2𝑜 ∈ ω) → ((𝐵 ∈ suc 𝐴 ∧ ∅ ∈ 2𝑜) ↔ (2𝑜 ·𝑜 𝐵) ∈ (2𝑜 ·𝑜 suc 𝐴)))
41	12, 40	mp3an3 1405	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ ω ∧ suc 𝐴 ∈ ω) → ((𝐵 ∈ suc 𝐴 ∧ ∅ ∈ 2𝑜) ↔ (2𝑜 ·𝑜 𝐵) ∈ (2𝑜 ·𝑜 suc 𝐴)))
42	39, 41	sylan2 490	. . . . . . . . . 10 ⊢ ((𝐵 ∈ ω ∧ 𝐴 ∈ ω) → ((𝐵 ∈ suc 𝐴 ∧ ∅ ∈ 2𝑜) ↔ (2𝑜 ·𝑜 𝐵) ∈ (2𝑜 ·𝑜 suc 𝐴)))
43	42	ancoms 468	. . . . . . . . 9 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐵 ∈ suc 𝐴 ∧ ∅ ∈ 2𝑜) ↔ (2𝑜 ·𝑜 𝐵) ∈ (2𝑜 ·𝑜 suc 𝐴)))
44	43	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)))
45	38, 44	mpbird 246	. . . . . . 7 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ suc (2𝑜 ·𝑜 𝐴) = (2𝑜 ·𝑜 𝐵)) → (𝐵 ∈ suc 𝐴 ∧ ∅ ∈ 2𝑜))
46	45	simpld 474	. . . . . 6 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ suc (2𝑜 ·𝑜 𝐴) = (2𝑜 ·𝑜 𝐵)) → 𝐵 ∈ suc 𝐴)
47	46	ex 449	. . . . 5 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (suc (2𝑜 ·𝑜 𝐴) = (2𝑜 ·𝑜 𝐵) → 𝐵 ∈ suc 𝐴))
48	17, 47	jcad 554	. . . 4 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (suc (2𝑜 ·𝑜 𝐴) = (2𝑜 ·𝑜 𝐵) → (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ suc 𝐴)))
49	48	3adant3 1074	. . 3 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 = (2𝑜 ·𝑜 𝐴)) → (suc (2𝑜 ·𝑜 𝐴) = (2𝑜 ·𝑜 𝐵) → (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ suc 𝐴)))
50	7, 49	sylbid 229	. 2 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 = (2𝑜 ·𝑜 𝐴)) → (suc 𝐶 = (2𝑜 ·𝑜 𝐵) → (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ suc 𝐴)))
51	4, 50	mtod 188	1 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 = (2𝑜 ·𝑜 𝐴)) → ¬ suc 𝐶 = (2𝑜 ·𝑜 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∅c0 3874  Oncon0 5640  suc csuc 5642  (class class class)co 6549  ωcom 6957  1_𝑜c1o 7440  2_𝑜c2o 7441   +_𝑜 coa 7444   ·_𝑜 comu 7445

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-omul 7452

This theorem is referenced by:  nneob  7619