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Theorem nneob 8016
Description: A natural number is even iff its successor is odd. (Contributed by NM, 26-Jan-2006.) (Revised by Mario Carneiro, 15-Nov-2014.)
Assertion
Ref Expression
nneob (𝐴 ∈ ω → (∃𝑥 ∈ ω 𝐴 = (2o ·o 𝑥) ↔ ¬ ∃𝑥 ∈ ω suc 𝐴 = (2o ·o 𝑥)))
Distinct variable group:   𝑥,𝐴

Proof of Theorem nneob
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 6930 . . . . 5 (𝑥 = 𝑦 → (2o ·o 𝑥) = (2o ·o 𝑦))
21eqeq2d 2787 . . . 4 (𝑥 = 𝑦 → (𝐴 = (2o ·o 𝑥) ↔ 𝐴 = (2o ·o 𝑦)))
32cbvrexv 3367 . . 3 (∃𝑥 ∈ ω 𝐴 = (2o ·o 𝑥) ↔ ∃𝑦 ∈ ω 𝐴 = (2o ·o 𝑦))
4 nnneo 8015 . . . . . . 7 ((𝑦 ∈ ω ∧ 𝑥 ∈ ω ∧ 𝐴 = (2o ·o 𝑦)) → ¬ suc 𝐴 = (2o ·o 𝑥))
543com23 1117 . . . . . 6 ((𝑦 ∈ ω ∧ 𝐴 = (2o ·o 𝑦) ∧ 𝑥 ∈ ω) → ¬ suc 𝐴 = (2o ·o 𝑥))
653expa 1108 . . . . 5 (((𝑦 ∈ ω ∧ 𝐴 = (2o ·o 𝑦)) ∧ 𝑥 ∈ ω) → ¬ suc 𝐴 = (2o ·o 𝑥))
76nrexdv 3181 . . . 4 ((𝑦 ∈ ω ∧ 𝐴 = (2o ·o 𝑦)) → ¬ ∃𝑥 ∈ ω suc 𝐴 = (2o ·o 𝑥))
87rexlimiva 3209 . . 3 (∃𝑦 ∈ ω 𝐴 = (2o ·o 𝑦) → ¬ ∃𝑥 ∈ ω suc 𝐴 = (2o ·o 𝑥))
93, 8sylbi 209 . 2 (∃𝑥 ∈ ω 𝐴 = (2o ·o 𝑥) → ¬ ∃𝑥 ∈ ω suc 𝐴 = (2o ·o 𝑥))
10 suceq 6041 . . . . . . 7 (𝑦 = ∅ → suc 𝑦 = suc ∅)
1110eqeq1d 2779 . . . . . 6 (𝑦 = ∅ → (suc 𝑦 = (2o ·o 𝑥) ↔ suc ∅ = (2o ·o 𝑥)))
1211rexbidv 3236 . . . . 5 (𝑦 = ∅ → (∃𝑥 ∈ ω suc 𝑦 = (2o ·o 𝑥) ↔ ∃𝑥 ∈ ω suc ∅ = (2o ·o 𝑥)))
1312notbid 310 . . . 4 (𝑦 = ∅ → (¬ ∃𝑥 ∈ ω suc 𝑦 = (2o ·o 𝑥) ↔ ¬ ∃𝑥 ∈ ω suc ∅ = (2o ·o 𝑥)))
14 eqeq1 2781 . . . . 5 (𝑦 = ∅ → (𝑦 = (2o ·o 𝑥) ↔ ∅ = (2o ·o 𝑥)))
1514rexbidv 3236 . . . 4 (𝑦 = ∅ → (∃𝑥 ∈ ω 𝑦 = (2o ·o 𝑥) ↔ ∃𝑥 ∈ ω ∅ = (2o ·o 𝑥)))
1613, 15imbi12d 336 . . 3 (𝑦 = ∅ → ((¬ ∃𝑥 ∈ ω suc 𝑦 = (2o ·o 𝑥) → ∃𝑥 ∈ ω 𝑦 = (2o ·o 𝑥)) ↔ (¬ ∃𝑥 ∈ ω suc ∅ = (2o ·o 𝑥) → ∃𝑥 ∈ ω ∅ = (2o ·o 𝑥))))
17 suceq 6041 . . . . . . 7 (𝑦 = 𝑧 → suc 𝑦 = suc 𝑧)
1817eqeq1d 2779 . . . . . 6 (𝑦 = 𝑧 → (suc 𝑦 = (2o ·o 𝑥) ↔ suc 𝑧 = (2o ·o 𝑥)))
1918rexbidv 3236 . . . . 5 (𝑦 = 𝑧 → (∃𝑥 ∈ ω suc 𝑦 = (2o ·o 𝑥) ↔ ∃𝑥 ∈ ω suc 𝑧 = (2o ·o 𝑥)))
2019notbid 310 . . . 4 (𝑦 = 𝑧 → (¬ ∃𝑥 ∈ ω suc 𝑦 = (2o ·o 𝑥) ↔ ¬ ∃𝑥 ∈ ω suc 𝑧 = (2o ·o 𝑥)))
21 eqeq1 2781 . . . . 5 (𝑦 = 𝑧 → (𝑦 = (2o ·o 𝑥) ↔ 𝑧 = (2o ·o 𝑥)))
2221rexbidv 3236 . . . 4 (𝑦 = 𝑧 → (∃𝑥 ∈ ω 𝑦 = (2o ·o 𝑥) ↔ ∃𝑥 ∈ ω 𝑧 = (2o ·o 𝑥)))
2320, 22imbi12d 336 . . 3 (𝑦 = 𝑧 → ((¬ ∃𝑥 ∈ ω suc 𝑦 = (2o ·o 𝑥) → ∃𝑥 ∈ ω 𝑦 = (2o ·o 𝑥)) ↔ (¬ ∃𝑥 ∈ ω suc 𝑧 = (2o ·o 𝑥) → ∃𝑥 ∈ ω 𝑧 = (2o ·o 𝑥))))
24 suceq 6041 . . . . . . 7 (𝑦 = suc 𝑧 → suc 𝑦 = suc suc 𝑧)
2524eqeq1d 2779 . . . . . 6 (𝑦 = suc 𝑧 → (suc 𝑦 = (2o ·o 𝑥) ↔ suc suc 𝑧 = (2o ·o 𝑥)))
2625rexbidv 3236 . . . . 5 (𝑦 = suc 𝑧 → (∃𝑥 ∈ ω suc 𝑦 = (2o ·o 𝑥) ↔ ∃𝑥 ∈ ω suc suc 𝑧 = (2o ·o 𝑥)))
2726notbid 310 . . . 4 (𝑦 = suc 𝑧 → (¬ ∃𝑥 ∈ ω suc 𝑦 = (2o ·o 𝑥) ↔ ¬ ∃𝑥 ∈ ω suc suc 𝑧 = (2o ·o 𝑥)))
28 eqeq1 2781 . . . . 5 (𝑦 = suc 𝑧 → (𝑦 = (2o ·o 𝑥) ↔ suc 𝑧 = (2o ·o 𝑥)))
2928rexbidv 3236 . . . 4 (𝑦 = suc 𝑧 → (∃𝑥 ∈ ω 𝑦 = (2o ·o 𝑥) ↔ ∃𝑥 ∈ ω suc 𝑧 = (2o ·o 𝑥)))
3027, 29imbi12d 336 . . 3 (𝑦 = suc 𝑧 → ((¬ ∃𝑥 ∈ ω suc 𝑦 = (2o ·o 𝑥) → ∃𝑥 ∈ ω 𝑦 = (2o ·o 𝑥)) ↔ (¬ ∃𝑥 ∈ ω suc suc 𝑧 = (2o ·o 𝑥) → ∃𝑥 ∈ ω suc 𝑧 = (2o ·o 𝑥))))
31 suceq 6041 . . . . . . 7 (𝑦 = 𝐴 → suc 𝑦 = suc 𝐴)
3231eqeq1d 2779 . . . . . 6 (𝑦 = 𝐴 → (suc 𝑦 = (2o ·o 𝑥) ↔ suc 𝐴 = (2o ·o 𝑥)))
3332rexbidv 3236 . . . . 5 (𝑦 = 𝐴 → (∃𝑥 ∈ ω suc 𝑦 = (2o ·o 𝑥) ↔ ∃𝑥 ∈ ω suc 𝐴 = (2o ·o 𝑥)))
3433notbid 310 . . . 4 (𝑦 = 𝐴 → (¬ ∃𝑥 ∈ ω suc 𝑦 = (2o ·o 𝑥) ↔ ¬ ∃𝑥 ∈ ω suc 𝐴 = (2o ·o 𝑥)))
35 eqeq1 2781 . . . . 5 (𝑦 = 𝐴 → (𝑦 = (2o ·o 𝑥) ↔ 𝐴 = (2o ·o 𝑥)))
3635rexbidv 3236 . . . 4 (𝑦 = 𝐴 → (∃𝑥 ∈ ω 𝑦 = (2o ·o 𝑥) ↔ ∃𝑥 ∈ ω 𝐴 = (2o ·o 𝑥)))
3734, 36imbi12d 336 . . 3 (𝑦 = 𝐴 → ((¬ ∃𝑥 ∈ ω suc 𝑦 = (2o ·o 𝑥) → ∃𝑥 ∈ ω 𝑦 = (2o ·o 𝑥)) ↔ (¬ ∃𝑥 ∈ ω suc 𝐴 = (2o ·o 𝑥) → ∃𝑥 ∈ ω 𝐴 = (2o ·o 𝑥))))
38 peano1 7363 . . . . 5 ∅ ∈ ω
39 eqid 2777 . . . . 5 ∅ = ∅
40 oveq2 6930 . . . . . . 7 (𝑥 = ∅ → (2o ·o 𝑥) = (2o ·o ∅))
41 2on 7852 . . . . . . . 8 2o ∈ On
42 om0 7881 . . . . . . . 8 (2o ∈ On → (2o ·o ∅) = ∅)
4341, 42ax-mp 5 . . . . . . 7 (2o ·o ∅) = ∅
4440, 43syl6eq 2829 . . . . . 6 (𝑥 = ∅ → (2o ·o 𝑥) = ∅)
4544rspceeqv 3528 . . . . 5 ((∅ ∈ ω ∧ ∅ = ∅) → ∃𝑥 ∈ ω ∅ = (2o ·o 𝑥))
4638, 39, 45mp2an 682 . . . 4 𝑥 ∈ ω ∅ = (2o ·o 𝑥)
4746a1i 11 . . 3 (¬ ∃𝑥 ∈ ω suc ∅ = (2o ·o 𝑥) → ∃𝑥 ∈ ω ∅ = (2o ·o 𝑥))
481eqeq2d 2787 . . . . . . 7 (𝑥 = 𝑦 → (𝑧 = (2o ·o 𝑥) ↔ 𝑧 = (2o ·o 𝑦)))
4948cbvrexv 3367 . . . . . 6 (∃𝑥 ∈ ω 𝑧 = (2o ·o 𝑥) ↔ ∃𝑦 ∈ ω 𝑧 = (2o ·o 𝑦))
50 peano2 7364 . . . . . . . . . 10 (𝑦 ∈ ω → suc 𝑦 ∈ ω)
51 2onn 8004 . . . . . . . . . . . 12 2o ∈ ω
52 nnmsuc 7971 . . . . . . . . . . . 12 ((2o ∈ ω ∧ 𝑦 ∈ ω) → (2o ·o suc 𝑦) = ((2o ·o 𝑦) +o 2o))
5351, 52mpan 680 . . . . . . . . . . 11 (𝑦 ∈ ω → (2o ·o suc 𝑦) = ((2o ·o 𝑦) +o 2o))
54 df-2o 7844 . . . . . . . . . . . . 13 2o = suc 1o
5554oveq2i 6933 . . . . . . . . . . . 12 ((2o ·o 𝑦) +o 2o) = ((2o ·o 𝑦) +o suc 1o)
56 nnmcl 7976 . . . . . . . . . . . . . 14 ((2o ∈ ω ∧ 𝑦 ∈ ω) → (2o ·o 𝑦) ∈ ω)
5751, 56mpan 680 . . . . . . . . . . . . 13 (𝑦 ∈ ω → (2o ·o 𝑦) ∈ ω)
58 1onn 8003 . . . . . . . . . . . . 13 1o ∈ ω
59 nnasuc 7970 . . . . . . . . . . . . 13 (((2o ·o 𝑦) ∈ ω ∧ 1o ∈ ω) → ((2o ·o 𝑦) +o suc 1o) = suc ((2o ·o 𝑦) +o 1o))
6057, 58, 59sylancl 580 . . . . . . . . . . . 12 (𝑦 ∈ ω → ((2o ·o 𝑦) +o suc 1o) = suc ((2o ·o 𝑦) +o 1o))
6155, 60syl5req 2826 . . . . . . . . . . 11 (𝑦 ∈ ω → suc ((2o ·o 𝑦) +o 1o) = ((2o ·o 𝑦) +o 2o))
62 nnon 7349 . . . . . . . . . . . 12 ((2o ·o 𝑦) ∈ ω → (2o ·o 𝑦) ∈ On)
63 oa1suc 7895 . . . . . . . . . . . 12 ((2o ·o 𝑦) ∈ On → ((2o ·o 𝑦) +o 1o) = suc (2o ·o 𝑦))
64 suceq 6041 . . . . . . . . . . . 12 (((2o ·o 𝑦) +o 1o) = suc (2o ·o 𝑦) → suc ((2o ·o 𝑦) +o 1o) = suc suc (2o ·o 𝑦))
6557, 62, 63, 644syl 19 . . . . . . . . . . 11 (𝑦 ∈ ω → suc ((2o ·o 𝑦) +o 1o) = suc suc (2o ·o 𝑦))
6653, 61, 653eqtr2rd 2820 . . . . . . . . . 10 (𝑦 ∈ ω → suc suc (2o ·o 𝑦) = (2o ·o suc 𝑦))
67 oveq2 6930 . . . . . . . . . . 11 (𝑥 = suc 𝑦 → (2o ·o 𝑥) = (2o ·o suc 𝑦))
6867rspceeqv 3528 . . . . . . . . . 10 ((suc 𝑦 ∈ ω ∧ suc suc (2o ·o 𝑦) = (2o ·o suc 𝑦)) → ∃𝑥 ∈ ω suc suc (2o ·o 𝑦) = (2o ·o 𝑥))
6950, 66, 68syl2anc 579 . . . . . . . . 9 (𝑦 ∈ ω → ∃𝑥 ∈ ω suc suc (2o ·o 𝑦) = (2o ·o 𝑥))
70 suceq 6041 . . . . . . . . . . . 12 (𝑧 = (2o ·o 𝑦) → suc 𝑧 = suc (2o ·o 𝑦))
71 suceq 6041 . . . . . . . . . . . 12 (suc 𝑧 = suc (2o ·o 𝑦) → suc suc 𝑧 = suc suc (2o ·o 𝑦))
7270, 71syl 17 . . . . . . . . . . 11 (𝑧 = (2o ·o 𝑦) → suc suc 𝑧 = suc suc (2o ·o 𝑦))
7372eqeq1d 2779 . . . . . . . . . 10 (𝑧 = (2o ·o 𝑦) → (suc suc 𝑧 = (2o ·o 𝑥) ↔ suc suc (2o ·o 𝑦) = (2o ·o 𝑥)))
7473rexbidv 3236 . . . . . . . . 9 (𝑧 = (2o ·o 𝑦) → (∃𝑥 ∈ ω suc suc 𝑧 = (2o ·o 𝑥) ↔ ∃𝑥 ∈ ω suc suc (2o ·o 𝑦) = (2o ·o 𝑥)))
7569, 74syl5ibrcom 239 . . . . . . . 8 (𝑦 ∈ ω → (𝑧 = (2o ·o 𝑦) → ∃𝑥 ∈ ω suc suc 𝑧 = (2o ·o 𝑥)))
7675rexlimiv 3208 . . . . . . 7 (∃𝑦 ∈ ω 𝑧 = (2o ·o 𝑦) → ∃𝑥 ∈ ω suc suc 𝑧 = (2o ·o 𝑥))
7776a1i 11 . . . . . 6 (𝑧 ∈ ω → (∃𝑦 ∈ ω 𝑧 = (2o ·o 𝑦) → ∃𝑥 ∈ ω suc suc 𝑧 = (2o ·o 𝑥)))
7849, 77syl5bi 234 . . . . 5 (𝑧 ∈ ω → (∃𝑥 ∈ ω 𝑧 = (2o ·o 𝑥) → ∃𝑥 ∈ ω suc suc 𝑧 = (2o ·o 𝑥)))
7978con3d 150 . . . 4 (𝑧 ∈ ω → (¬ ∃𝑥 ∈ ω suc suc 𝑧 = (2o ·o 𝑥) → ¬ ∃𝑥 ∈ ω 𝑧 = (2o ·o 𝑥)))
80 con1 146 . . . 4 ((¬ ∃𝑥 ∈ ω suc 𝑧 = (2o ·o 𝑥) → ∃𝑥 ∈ ω 𝑧 = (2o ·o 𝑥)) → (¬ ∃𝑥 ∈ ω 𝑧 = (2o ·o 𝑥) → ∃𝑥 ∈ ω suc 𝑧 = (2o ·o 𝑥)))
8179, 80syl9 77 . . 3 (𝑧 ∈ ω → ((¬ ∃𝑥 ∈ ω suc 𝑧 = (2o ·o 𝑥) → ∃𝑥 ∈ ω 𝑧 = (2o ·o 𝑥)) → (¬ ∃𝑥 ∈ ω suc suc 𝑧 = (2o ·o 𝑥) → ∃𝑥 ∈ ω suc 𝑧 = (2o ·o 𝑥))))
8216, 23, 30, 37, 47, 81finds 7370 . 2 (𝐴 ∈ ω → (¬ ∃𝑥 ∈ ω suc 𝐴 = (2o ·o 𝑥) → ∃𝑥 ∈ ω 𝐴 = (2o ·o 𝑥)))
839, 82impbid2 218 1 (𝐴 ∈ ω → (∃𝑥 ∈ ω 𝐴 = (2o ·o 𝑥) ↔ ¬ ∃𝑥 ∈ ω suc 𝐴 = (2o ·o 𝑥)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 198  wa 386   = wceq 1601  wcel 2106  wrex 3090  c0 4140  Oncon0 5976  suc csuc 5978  (class class class)co 6922  ωcom 7343  1oc1o 7836  2oc2o 7837   +o coa 7840   ·o comu 7841
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-8 2108  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2550  df-eu 2586  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ne 2969  df-ral 3094  df-rex 3095  df-reu 3096  df-rab 3098  df-v 3399  df-sbc 3652  df-csb 3751  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-pss 3807  df-nul 4141  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-tp 4402  df-op 4404  df-uni 4672  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-tr 4988  df-id 5261  df-eprel 5266  df-po 5274  df-so 5275  df-fr 5314  df-we 5316  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-pred 5933  df-ord 5979  df-on 5980  df-lim 5981  df-suc 5982  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-om 7344  df-wrecs 7689  df-recs 7751  df-rdg 7789  df-1o 7843  df-2o 7844  df-oadd 7847  df-omul 7848
This theorem is referenced by:  fin1a2lem5  9561
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