Theorem nnneo 8692

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 6480	. . . 4 ⊢ (𝐴 ∈ On → ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ suc 𝐴))
3	1, 2	syl 17	. . 3 ⊢ (𝐴 ∈ ω → ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ suc 𝐴))
4	3	3ad2ant1 1132	. 2 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 = (2o ·o 𝐴)) → ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ suc 𝐴))
5		suceq 6452	. . . . 5 ⊢ (𝐶 = (2o ·o 𝐴) → suc 𝐶 = suc (2o ·o 𝐴))
6	5	eqeq1d 2737	. . . 4 ⊢ (𝐶 = (2o ·o 𝐴) → (suc 𝐶 = (2o ·o 𝐵) ↔ suc (2o ·o 𝐴) = (2o ·o 𝐵)))
7	6	3ad2ant3 1134	. . 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 6468	. . . . . . 7 ⊢ (2o ·o 𝐴) ∈ suc (2o ·o 𝐴)
10		eleq2 2828	. . . . . . 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 8679	. . . . . . . 8 ⊢ 2o ∈ ω
13		nnmord 8669	. . . . . . . 8 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 2o ∈ ω) → ((𝐴 ∈ 𝐵 ∧ ∅ ∈ 2o) ↔ (2o ·o 𝐴) ∈ (2o ·o 𝐵)))
14	12, 13	mp3an3 1449	. . . . . . 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 8649	. . . . . . . . . . . . 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 8568	. . . . . . . . . . . 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 8515	. . . . . . . . . . . . . . 15 ⊢ 1o ∈ V
25	24	sucid 6468	. . . . . . . . . . . . . 14 ⊢ 1o ∈ suc 1o
26		df-2o 8506	. . . . . . . . . . . . . 14 ⊢ 2o = suc 1o
27	25, 26	eleqtrri 2838	. . . . . . . . . . . . 13 ⊢ 1o ∈ 2o
28		1onn 8677	. . . . . . . . . . . . . 14 ⊢ 1o ∈ ω
29		nnaord 8656	. . . . . . . . . . . . . 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 1464	. . . . . . . . . . . . 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 8644	. . . . . . . . . . . . 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 2842	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ω → ((2o ·o 𝐴) +o 1o) ∈ (2o ·o suc 𝐴))
35	23, 34	eqeltrrd 2840	. . . . . . . . . 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 2840	. . . . . . . 8 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ suc (2o ·o 𝐴) = (2o ·o 𝐵)) → (2o ·o 𝐵) ∈ (2o ·o suc 𝐴))
38		peano2 7913	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ω → suc 𝐴 ∈ ω)
39		nnmord 8669	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ ω ∧ suc 𝐴 ∈ ω ∧ 2o ∈ ω) → ((𝐵 ∈ suc 𝐴 ∧ ∅ ∈ 2o) ↔ (2o ·o 𝐵) ∈ (2o ·o suc 𝐴)))
40	12, 39	mp3an3 1449	. . . . . . . . . . 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 1131	. . 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 1086   = wceq 1537   ∈ wcel 2106  ∅c0 4339  Oncon0 6386  suc csuc 6388  (class class class)co 7431  ωcom 7887  1_oc1o 8498  2_oc2o 8499   +_o coa 8502   ·_o comu 8503

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-2o 8506  df-oadd 8509  df-omul 8510

This theorem is referenced by:  nneob  8693