Theorem nnneo 8647

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 7874	. . . 4 ⊢ (𝐴 ∈ ω → 𝐴 ∈ On)
2		onnbtwn 6461	. . . 4 ⊢ (𝐴 ∈ On → ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ suc 𝐴))
3	1, 2	syl 18	. . 3 ⊢ (𝐴 ∈ ω → ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ suc 𝐴))
4	3	3ad2ant1 1151	. 2 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 = (2o ·o 𝐴)) → ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ suc 𝐴))
5		suceq 6433	. . . . 5 ⊢ (𝐶 = (2o ·o 𝐴) → suc 𝐶 = suc (2o ·o 𝐴))
6	5	eqeq1d 2767	. . . 4 ⊢ (𝐶 = (2o ·o 𝐴) → (suc 𝐶 = (2o ·o 𝐵) ↔ suc (2o ·o 𝐴) = (2o ·o 𝐵)))
7	6	3ad2ant3 1153	. . 3 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 = (2o ·o 𝐴)) → (suc 𝐶 = (2o ·o 𝐵) ↔ suc (2o ·o 𝐴) = (2o ·o 𝐵)))
8		ovex 7452	. . . . . . . 8 ⊢ (2o ·o 𝐴) ∈ V
9	8	sucid 6449	. . . . . . 7 ⊢ (2o ·o 𝐴) ∈ suc (2o ·o 𝐴)
10		eleq2 2854	. . . . . . 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 236	. . . . . 6 ⊢ (suc (2o ·o 𝐴) = (2o ·o 𝐵) → (2o ·o 𝐴) ∈ (2o ·o 𝐵))
12		2onn 8634	. . . . . . . 8 ⊢ 2o ∈ ω
13		nnmord 8624	. . . . . . . 8 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 2o ∈ ω) → ((𝐴 ∈ 𝐵 ∧ ∅ ∈ 2o) ↔ (2o ·o 𝐴) ∈ (2o ·o 𝐵)))
14	12, 13	mp3an3 1479	. . . . . . 7 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐴 ∈ 𝐵 ∧ ∅ ∈ 2o) ↔ (2o ·o 𝐴) ∈ (2o ·o 𝐵)))
15		simpl 488	. . . . . . 7 ⊢ ((𝐴 ∈ 𝐵 ∧ ∅ ∈ 2o) → 𝐴 ∈ 𝐵)
16	14, 15	biimtrrdi 257	. . . . . 6 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((2o ·o 𝐴) ∈ (2o ·o 𝐵) → 𝐴 ∈ 𝐵))
17	11, 16	syl5 35	. . . . 5 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (suc (2o ·o 𝐴) = (2o ·o 𝐵) → 𝐴 ∈ 𝐵))
18		simpr 490	. . . . . . . . 9 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ suc (2o ·o 𝐴) = (2o ·o 𝐵)) → suc (2o ·o 𝐴) = (2o ·o 𝐵))
19		nnmcl 8604	. . . . . . . . . . . . 13 ⊢ ((2o ∈ ω ∧ 𝐴 ∈ ω) → (2o ·o 𝐴) ∈ ω)
20	12, 19	mpan 703	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ω → (2o ·o 𝐴) ∈ ω)
21		nnon 7874	. . . . . . . . . . . 12 ⊢ ((2o ·o 𝐴) ∈ ω → (2o ·o 𝐴) ∈ On)
22		oa1suc 8522	. . . . . . . . . . . 12 ⊢ ((2o ·o 𝐴) ∈ On → ((2o ·o 𝐴) +o 1o) = suc (2o ·o 𝐴))
23	20, 21, 22	3syl 19	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ω → ((2o ·o 𝐴) +o 1o) = suc (2o ·o 𝐴))
24		1oex 8469	. . . . . . . . . . . . . . 15 ⊢ 1o ∈ V
25	24	sucid 6449	. . . . . . . . . . . . . 14 ⊢ 1o ∈ suc 1o
26		df-2o 8460	. . . . . . . . . . . . . 14 ⊢ 2o = suc 1o
27	25, 26	eleqtrri 2864	. . . . . . . . . . . . 13 ⊢ 1o ∈ 2o
28		1onn 8632	. . . . . . . . . . . . . 14 ⊢ 1o ∈ ω
29		nnaord 8611	. . . . . . . . . . . . . 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 1494	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ω → (1o ∈ 2o ↔ ((2o ·o 𝐴) +o 1o) ∈ ((2o ·o 𝐴) +o 2o)))
31	27, 30	mpbii 236	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ω → ((2o ·o 𝐴) +o 1o) ∈ ((2o ·o 𝐴) +o 2o))
32		nnmsuc 8599	. . . . . . . . . . . . 13 ⊢ ((2o ∈ ω ∧ 𝐴 ∈ ω) → (2o ·o suc 𝐴) = ((2o ·o 𝐴) +o 2o))
33	12, 32	mpan 703	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ω → (2o ·o suc 𝐴) = ((2o ·o 𝐴) +o 2o))
34	31, 33	eleqtrrd 2868	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ω → ((2o ·o 𝐴) +o 1o) ∈ (2o ·o suc 𝐴))
35	23, 34	eqeltrrd 2866	. . . . . . . . . 10 ⊢ (𝐴 ∈ ω → suc (2o ·o 𝐴) ∈ (2o ·o suc 𝐴))
36	35	ad2antrr 739	. . . . . . . . 9 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ suc (2o ·o 𝐴) = (2o ·o 𝐵)) → suc (2o ·o 𝐴) ∈ (2o ·o suc 𝐴))
37	18, 36	eqeltrrd 2866	. . . . . . . 8 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ suc (2o ·o 𝐴) = (2o ·o 𝐵)) → (2o ·o 𝐵) ∈ (2o ·o suc 𝐴))
38		peano2 7892	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ω → suc 𝐴 ∈ ω)
39		nnmord 8624	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ ω ∧ suc 𝐴 ∈ ω ∧ 2o ∈ ω) → ((𝐵 ∈ suc 𝐴 ∧ ∅ ∈ 2o) ↔ (2o ·o 𝐵) ∈ (2o ·o suc 𝐴)))
40	12, 39	mp3an3 1479	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ ω ∧ suc 𝐴 ∈ ω) → ((𝐵 ∈ suc 𝐴 ∧ ∅ ∈ 2o) ↔ (2o ·o 𝐵) ∈ (2o ·o suc 𝐴)))
41	38, 40	sylan2 605	. . . . . . . . . 10 ⊢ ((𝐵 ∈ ω ∧ 𝐴 ∈ ω) → ((𝐵 ∈ suc 𝐴 ∧ ∅ ∈ 2o) ↔ (2o ·o 𝐵) ∈ (2o ·o suc 𝐴)))
42	41	ancoms 464	. . . . . . . . 9 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐵 ∈ suc 𝐴 ∧ ∅ ∈ 2o) ↔ (2o ·o 𝐵) ∈ (2o ·o suc 𝐴)))
43	42	adantr 486	. . . . . . . 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 260	. . . . . . 7 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ suc (2o ·o 𝐴) = (2o ·o 𝐵)) → (𝐵 ∈ suc 𝐴 ∧ ∅ ∈ 2o))
45	44	simpld 500	. . . . . 6 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ suc (2o ·o 𝐴) = (2o ·o 𝐵)) → 𝐵 ∈ suc 𝐴)
46	45	ex 418	. . . . 5 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (suc (2o ·o 𝐴) = (2o ·o 𝐵) → 𝐵 ∈ suc 𝐴))
47	17, 46	jcad 522	. . . 4 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (suc (2o ·o 𝐴) = (2o ·o 𝐵) → (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ suc 𝐴)))
48	47	3adant3 1150	. . 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 243	. 2 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 = (2o ·o 𝐴)) → (suc 𝐶 = (2o ·o 𝐵) → (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ suc 𝐴)))
50	4, 49	mtod 201	1 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 = (2o ·o 𝐴)) → ¬ suc 𝐶 = (2o ·o 𝐵))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146  ∅c0 4286  Oncon0 6364  suc csuc 6366  (class class class)co 7419  ωcom 7868  1_oc1o 8452  2_oc2o 8453   +_o coa 8456   ·_o comu 8457

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pr 5406  ax-un 7742

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6306  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7422  df-oprab 7423  df-mpo 7424  df-om 7869  df-2nd 7993  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-rdg 8403  df-1o 8459  df-2o 8460  df-oadd 8463  df-omul 8464

This theorem is used by:  nneob  8648