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Theorem nnneo 7677
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
StepHypRef Expression
1 nnon 7019 . . . 4 (𝐴 ∈ ω → 𝐴 ∈ On)
2 onnbtwn 5780 . . . 4 (𝐴 ∈ On → ¬ (𝐴𝐵𝐵 ∈ suc 𝐴))
31, 2syl 17 . . 3 (𝐴 ∈ ω → ¬ (𝐴𝐵𝐵 ∈ suc 𝐴))
433ad2ant1 1080 . 2 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 = (2𝑜 ·𝑜 𝐴)) → ¬ (𝐴𝐵𝐵 ∈ suc 𝐴))
5 suceq 5752 . . . . 5 (𝐶 = (2𝑜 ·𝑜 𝐴) → suc 𝐶 = suc (2𝑜 ·𝑜 𝐴))
65eqeq1d 2628 . . . 4 (𝐶 = (2𝑜 ·𝑜 𝐴) → (suc 𝐶 = (2𝑜 ·𝑜 𝐵) ↔ suc (2𝑜 ·𝑜 𝐴) = (2𝑜 ·𝑜 𝐵)))
763ad2ant3 1082 . . 3 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 = (2𝑜 ·𝑜 𝐴)) → (suc 𝐶 = (2𝑜 ·𝑜 𝐵) ↔ suc (2𝑜 ·𝑜 𝐴) = (2𝑜 ·𝑜 𝐵)))
8 ovex 6633 . . . . . . . 8 (2𝑜 ·𝑜 𝐴) ∈ V
98sucid 5766 . . . . . . 7 (2𝑜 ·𝑜 𝐴) ∈ suc (2𝑜 ·𝑜 𝐴)
10 eleq2 2693 . . . . . . 7 (suc (2𝑜 ·𝑜 𝐴) = (2𝑜 ·𝑜 𝐵) → ((2𝑜 ·𝑜 𝐴) ∈ suc (2𝑜 ·𝑜 𝐴) ↔ (2𝑜 ·𝑜 𝐴) ∈ (2𝑜 ·𝑜 𝐵)))
119, 10mpbii 223 . . . . . 6 (suc (2𝑜 ·𝑜 𝐴) = (2𝑜 ·𝑜 𝐵) → (2𝑜 ·𝑜 𝐴) ∈ (2𝑜 ·𝑜 𝐵))
12 2onn 7666 . . . . . . . 8 2𝑜 ∈ ω
13 nnmord 7658 . . . . . . . 8 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 2𝑜 ∈ ω) → ((𝐴𝐵 ∧ ∅ ∈ 2𝑜) ↔ (2𝑜 ·𝑜 𝐴) ∈ (2𝑜 ·𝑜 𝐵)))
1412, 13mp3an3 1410 . . . . . . 7 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐴𝐵 ∧ ∅ ∈ 2𝑜) ↔ (2𝑜 ·𝑜 𝐴) ∈ (2𝑜 ·𝑜 𝐵)))
15 simpl 473 . . . . . . 7 ((𝐴𝐵 ∧ ∅ ∈ 2𝑜) → 𝐴𝐵)
1614, 15syl6bir 244 . . . . . 6 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((2𝑜 ·𝑜 𝐴) ∈ (2𝑜 ·𝑜 𝐵) → 𝐴𝐵))
1711, 16syl5 34 . . . . 5 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (suc (2𝑜 ·𝑜 𝐴) = (2𝑜 ·𝑜 𝐵) → 𝐴𝐵))
18 simpr 477 . . . . . . . . 9 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ suc (2𝑜 ·𝑜 𝐴) = (2𝑜 ·𝑜 𝐵)) → suc (2𝑜 ·𝑜 𝐴) = (2𝑜 ·𝑜 𝐵))
19 nnmcl 7638 . . . . . . . . . . . . 13 ((2𝑜 ∈ ω ∧ 𝐴 ∈ ω) → (2𝑜 ·𝑜 𝐴) ∈ ω)
2012, 19mpan 705 . . . . . . . . . . . 12 (𝐴 ∈ ω → (2𝑜 ·𝑜 𝐴) ∈ ω)
21 nnon 7019 . . . . . . . . . . . 12 ((2𝑜 ·𝑜 𝐴) ∈ ω → (2𝑜 ·𝑜 𝐴) ∈ On)
22 oa1suc 7557 . . . . . . . . . . . 12 ((2𝑜 ·𝑜 𝐴) ∈ On → ((2𝑜 ·𝑜 𝐴) +𝑜 1𝑜) = suc (2𝑜 ·𝑜 𝐴))
2320, 21, 223syl 18 . . . . . . . . . . 11 (𝐴 ∈ ω → ((2𝑜 ·𝑜 𝐴) +𝑜 1𝑜) = suc (2𝑜 ·𝑜 𝐴))
24 1onn 7665 . . . . . . . . . . . . . . . 16 1𝑜 ∈ ω
2524elexi 3204 . . . . . . . . . . . . . . 15 1𝑜 ∈ V
2625sucid 5766 . . . . . . . . . . . . . 14 1𝑜 ∈ suc 1𝑜
27 df-2o 7507 . . . . . . . . . . . . . 14 2𝑜 = suc 1𝑜
2826, 27eleqtrri 2703 . . . . . . . . . . . . 13 1𝑜 ∈ 2𝑜
29 nnaord 7645 . . . . . . . . . . . . . . 15 ((1𝑜 ∈ ω ∧ 2𝑜 ∈ ω ∧ (2𝑜 ·𝑜 𝐴) ∈ ω) → (1𝑜 ∈ 2𝑜 ↔ ((2𝑜 ·𝑜 𝐴) +𝑜 1𝑜) ∈ ((2𝑜 ·𝑜 𝐴) +𝑜 2𝑜)))
3024, 12, 29mp3an12 1411 . . . . . . . . . . . . . 14 ((2𝑜 ·𝑜 𝐴) ∈ ω → (1𝑜 ∈ 2𝑜 ↔ ((2𝑜 ·𝑜 𝐴) +𝑜 1𝑜) ∈ ((2𝑜 ·𝑜 𝐴) +𝑜 2𝑜)))
3120, 30syl 17 . . . . . . . . . . . . 13 (𝐴 ∈ ω → (1𝑜 ∈ 2𝑜 ↔ ((2𝑜 ·𝑜 𝐴) +𝑜 1𝑜) ∈ ((2𝑜 ·𝑜 𝐴) +𝑜 2𝑜)))
3228, 31mpbii 223 . . . . . . . . . . . 12 (𝐴 ∈ ω → ((2𝑜 ·𝑜 𝐴) +𝑜 1𝑜) ∈ ((2𝑜 ·𝑜 𝐴) +𝑜 2𝑜))
33 nnmsuc 7633 . . . . . . . . . . . . 13 ((2𝑜 ∈ ω ∧ 𝐴 ∈ ω) → (2𝑜 ·𝑜 suc 𝐴) = ((2𝑜 ·𝑜 𝐴) +𝑜 2𝑜))
3412, 33mpan 705 . . . . . . . . . . . 12 (𝐴 ∈ ω → (2𝑜 ·𝑜 suc 𝐴) = ((2𝑜 ·𝑜 𝐴) +𝑜 2𝑜))
3532, 34eleqtrrd 2707 . . . . . . . . . . 11 (𝐴 ∈ ω → ((2𝑜 ·𝑜 𝐴) +𝑜 1𝑜) ∈ (2𝑜 ·𝑜 suc 𝐴))
3623, 35eqeltrrd 2705 . . . . . . . . . 10 (𝐴 ∈ ω → suc (2𝑜 ·𝑜 𝐴) ∈ (2𝑜 ·𝑜 suc 𝐴))
3736ad2antrr 761 . . . . . . . . 9 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ suc (2𝑜 ·𝑜 𝐴) = (2𝑜 ·𝑜 𝐵)) → suc (2𝑜 ·𝑜 𝐴) ∈ (2𝑜 ·𝑜 suc 𝐴))
3818, 37eqeltrrd 2705 . . . . . . . 8 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ suc (2𝑜 ·𝑜 𝐴) = (2𝑜 ·𝑜 𝐵)) → (2𝑜 ·𝑜 𝐵) ∈ (2𝑜 ·𝑜 suc 𝐴))
39 peano2 7034 . . . . . . . . . . 11 (𝐴 ∈ ω → suc 𝐴 ∈ ω)
40 nnmord 7658 . . . . . . . . . . . 12 ((𝐵 ∈ ω ∧ suc 𝐴 ∈ ω ∧ 2𝑜 ∈ ω) → ((𝐵 ∈ suc 𝐴 ∧ ∅ ∈ 2𝑜) ↔ (2𝑜 ·𝑜 𝐵) ∈ (2𝑜 ·𝑜 suc 𝐴)))
4112, 40mp3an3 1410 . . . . . . . . . . 11 ((𝐵 ∈ ω ∧ suc 𝐴 ∈ ω) → ((𝐵 ∈ suc 𝐴 ∧ ∅ ∈ 2𝑜) ↔ (2𝑜 ·𝑜 𝐵) ∈ (2𝑜 ·𝑜 suc 𝐴)))
4239, 41sylan2 491 . . . . . . . . . 10 ((𝐵 ∈ ω ∧ 𝐴 ∈ ω) → ((𝐵 ∈ suc 𝐴 ∧ ∅ ∈ 2𝑜) ↔ (2𝑜 ·𝑜 𝐵) ∈ (2𝑜 ·𝑜 suc 𝐴)))
4342ancoms 469 . . . . . . . . 9 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐵 ∈ suc 𝐴 ∧ ∅ ∈ 2𝑜) ↔ (2𝑜 ·𝑜 𝐵) ∈ (2𝑜 ·𝑜 suc 𝐴)))
4443adantr 481 . . . . . . . 8 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ suc (2𝑜 ·𝑜 𝐴) = (2𝑜 ·𝑜 𝐵)) → ((𝐵 ∈ suc 𝐴 ∧ ∅ ∈ 2𝑜) ↔ (2𝑜 ·𝑜 𝐵) ∈ (2𝑜 ·𝑜 suc 𝐴)))
4538, 44mpbird 247 . . . . . . 7 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ suc (2𝑜 ·𝑜 𝐴) = (2𝑜 ·𝑜 𝐵)) → (𝐵 ∈ suc 𝐴 ∧ ∅ ∈ 2𝑜))
4645simpld 475 . . . . . 6 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ suc (2𝑜 ·𝑜 𝐴) = (2𝑜 ·𝑜 𝐵)) → 𝐵 ∈ suc 𝐴)
4746ex 450 . . . . 5 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (suc (2𝑜 ·𝑜 𝐴) = (2𝑜 ·𝑜 𝐵) → 𝐵 ∈ suc 𝐴))
4817, 47jcad 555 . . . 4 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (suc (2𝑜 ·𝑜 𝐴) = (2𝑜 ·𝑜 𝐵) → (𝐴𝐵𝐵 ∈ suc 𝐴)))
49483adant3 1079 . . 3 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 = (2𝑜 ·𝑜 𝐴)) → (suc (2𝑜 ·𝑜 𝐴) = (2𝑜 ·𝑜 𝐵) → (𝐴𝐵𝐵 ∈ suc 𝐴)))
507, 49sylbid 230 . 2 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 = (2𝑜 ·𝑜 𝐴)) → (suc 𝐶 = (2𝑜 ·𝑜 𝐵) → (𝐴𝐵𝐵 ∈ suc 𝐴)))
514, 50mtod 189 1 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 = (2𝑜 ·𝑜 𝐴)) → ¬ suc 𝐶 = (2𝑜 ·𝑜 𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1992  c0 3896  Oncon0 5685  suc csuc 5687  (class class class)co 6605  ωcom 7013  1𝑜c1o 7499  2𝑜c2o 7500   +𝑜 coa 7503   ·𝑜 comu 7504
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5642  df-ord 5688  df-on 5689  df-lim 5690  df-suc 5691  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-om 7014  df-1st 7116  df-2nd 7117  df-wrecs 7353  df-recs 7414  df-rdg 7452  df-1o 7506  df-2o 7507  df-oadd 7510  df-omul 7511
This theorem is referenced by:  nneob  7678
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