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Theorem nnneo 7716
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 7056 . . . 4 (𝐴 ∈ ω → 𝐴 ∈ On)
2 onnbtwn 5806 . . . 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 5778 . . . . 5 (𝐶 = (2𝑜 ·𝑜 𝐴) → suc 𝐶 = suc (2𝑜 ·𝑜 𝐴))
65eqeq1d 2622 . . . 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 6663 . . . . . . . 8 (2𝑜 ·𝑜 𝐴) ∈ V
98sucid 5792 . . . . . . 7 (2𝑜 ·𝑜 𝐴) ∈ suc (2𝑜 ·𝑜 𝐴)
10 eleq2 2688 . . . . . . 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 7705 . . . . . . . 8 2𝑜 ∈ ω
13 nnmord 7697 . . . . . . . 8 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 2𝑜 ∈ ω) → ((𝐴𝐵 ∧ ∅ ∈ 2𝑜) ↔ (2𝑜 ·𝑜 𝐴) ∈ (2𝑜 ·𝑜 𝐵)))
1412, 13mp3an3 1411 . . . . . . 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 7677 . . . . . . . . . . . . 13 ((2𝑜 ∈ ω ∧ 𝐴 ∈ ω) → (2𝑜 ·𝑜 𝐴) ∈ ω)
2012, 19mpan 705 . . . . . . . . . . . 12 (𝐴 ∈ ω → (2𝑜 ·𝑜 𝐴) ∈ ω)
21 nnon 7056 . . . . . . . . . . . 12 ((2𝑜 ·𝑜 𝐴) ∈ ω → (2𝑜 ·𝑜 𝐴) ∈ On)
22 oa1suc 7596 . . . . . . . . . . . 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 7704 . . . . . . . . . . . . . . . 16 1𝑜 ∈ ω
2524elexi 3208 . . . . . . . . . . . . . . 15 1𝑜 ∈ V
2625sucid 5792 . . . . . . . . . . . . . 14 1𝑜 ∈ suc 1𝑜
27 df-2o 7546 . . . . . . . . . . . . . 14 2𝑜 = suc 1𝑜
2826, 27eleqtrri 2698 . . . . . . . . . . . . 13 1𝑜 ∈ 2𝑜
29 nnaord 7684 . . . . . . . . . . . . . . 15 ((1𝑜 ∈ ω ∧ 2𝑜 ∈ ω ∧ (2𝑜 ·𝑜 𝐴) ∈ ω) → (1𝑜 ∈ 2𝑜 ↔ ((2𝑜 ·𝑜 𝐴) +𝑜 1𝑜) ∈ ((2𝑜 ·𝑜 𝐴) +𝑜 2𝑜)))
3024, 12, 29mp3an12 1412 . . . . . . . . . . . . . 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 7672 . . . . . . . . . . . . 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 2702 . . . . . . . . . . 11 (𝐴 ∈ ω → ((2𝑜 ·𝑜 𝐴) +𝑜 1𝑜) ∈ (2𝑜 ·𝑜 suc 𝐴))
3623, 35eqeltrrd 2700 . . . . . . . . . 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 2700 . . . . . . . 8 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ suc (2𝑜 ·𝑜 𝐴) = (2𝑜 ·𝑜 𝐵)) → (2𝑜 ·𝑜 𝐵) ∈ (2𝑜 ·𝑜 suc 𝐴))
39 peano2 7071 . . . . . . . . . . 11 (𝐴 ∈ ω → suc 𝐴 ∈ ω)
40 nnmord 7697 . . . . . . . . . . . 12 ((𝐵 ∈ ω ∧ suc 𝐴 ∈ ω ∧ 2𝑜 ∈ ω) → ((𝐵 ∈ suc 𝐴 ∧ ∅ ∈ 2𝑜) ↔ (2𝑜 ·𝑜 𝐵) ∈ (2𝑜 ·𝑜 suc 𝐴)))
4112, 40mp3an3 1411 . . . . . . . . . . 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 1481  wcel 1988  c0 3907  Oncon0 5711  suc csuc 5713  (class class class)co 6635  ωcom 7050  1𝑜c1o 7538  2𝑜c2o 7539   +𝑜 coa 7542   ·𝑜 comu 7543
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-pred 5668  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-om 7051  df-1st 7153  df-2nd 7154  df-wrecs 7392  df-recs 7453  df-rdg 7491  df-1o 7545  df-2o 7546  df-oadd 7549  df-omul 7550
This theorem is referenced by:  nneob  7717
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