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Theorem nneob 8275
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 7157 . . . . 5 (𝑥 = 𝑦 → (2o ·o 𝑥) = (2o ·o 𝑦))
21eqeq2d 2835 . . . 4 (𝑥 = 𝑦 → (𝐴 = (2o ·o 𝑥) ↔ 𝐴 = (2o ·o 𝑦)))
32cbvrexvw 3435 . . 3 (∃𝑥 ∈ ω 𝐴 = (2o ·o 𝑥) ↔ ∃𝑦 ∈ ω 𝐴 = (2o ·o 𝑦))
4 nnneo 8274 . . . . . . 7 ((𝑦 ∈ ω ∧ 𝑥 ∈ ω ∧ 𝐴 = (2o ·o 𝑦)) → ¬ suc 𝐴 = (2o ·o 𝑥))
543com23 1123 . . . . . 6 ((𝑦 ∈ ω ∧ 𝐴 = (2o ·o 𝑦) ∧ 𝑥 ∈ ω) → ¬ suc 𝐴 = (2o ·o 𝑥))
653expa 1115 . . . . 5 (((𝑦 ∈ ω ∧ 𝐴 = (2o ·o 𝑦)) ∧ 𝑥 ∈ ω) → ¬ suc 𝐴 = (2o ·o 𝑥))
76nrexdv 3262 . . . 4 ((𝑦 ∈ ω ∧ 𝐴 = (2o ·o 𝑦)) → ¬ ∃𝑥 ∈ ω suc 𝐴 = (2o ·o 𝑥))
87rexlimiva 3273 . . 3 (∃𝑦 ∈ ω 𝐴 = (2o ·o 𝑦) → ¬ ∃𝑥 ∈ ω suc 𝐴 = (2o ·o 𝑥))
93, 8sylbi 220 . 2 (∃𝑥 ∈ ω 𝐴 = (2o ·o 𝑥) → ¬ ∃𝑥 ∈ ω suc 𝐴 = (2o ·o 𝑥))
10 suceq 6243 . . . . . . 7 (𝑦 = ∅ → suc 𝑦 = suc ∅)
1110eqeq1d 2826 . . . . . 6 (𝑦 = ∅ → (suc 𝑦 = (2o ·o 𝑥) ↔ suc ∅ = (2o ·o 𝑥)))
1211rexbidv 3289 . . . . 5 (𝑦 = ∅ → (∃𝑥 ∈ ω suc 𝑦 = (2o ·o 𝑥) ↔ ∃𝑥 ∈ ω suc ∅ = (2o ·o 𝑥)))
1312notbid 321 . . . 4 (𝑦 = ∅ → (¬ ∃𝑥 ∈ ω suc 𝑦 = (2o ·o 𝑥) ↔ ¬ ∃𝑥 ∈ ω suc ∅ = (2o ·o 𝑥)))
14 eqeq1 2828 . . . . 5 (𝑦 = ∅ → (𝑦 = (2o ·o 𝑥) ↔ ∅ = (2o ·o 𝑥)))
1514rexbidv 3289 . . . 4 (𝑦 = ∅ → (∃𝑥 ∈ ω 𝑦 = (2o ·o 𝑥) ↔ ∃𝑥 ∈ ω ∅ = (2o ·o 𝑥)))
1613, 15imbi12d 348 . . 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 6243 . . . . . . 7 (𝑦 = 𝑧 → suc 𝑦 = suc 𝑧)
1817eqeq1d 2826 . . . . . 6 (𝑦 = 𝑧 → (suc 𝑦 = (2o ·o 𝑥) ↔ suc 𝑧 = (2o ·o 𝑥)))
1918rexbidv 3289 . . . . 5 (𝑦 = 𝑧 → (∃𝑥 ∈ ω suc 𝑦 = (2o ·o 𝑥) ↔ ∃𝑥 ∈ ω suc 𝑧 = (2o ·o 𝑥)))
2019notbid 321 . . . 4 (𝑦 = 𝑧 → (¬ ∃𝑥 ∈ ω suc 𝑦 = (2o ·o 𝑥) ↔ ¬ ∃𝑥 ∈ ω suc 𝑧 = (2o ·o 𝑥)))
21 eqeq1 2828 . . . . 5 (𝑦 = 𝑧 → (𝑦 = (2o ·o 𝑥) ↔ 𝑧 = (2o ·o 𝑥)))
2221rexbidv 3289 . . . 4 (𝑦 = 𝑧 → (∃𝑥 ∈ ω 𝑦 = (2o ·o 𝑥) ↔ ∃𝑥 ∈ ω 𝑧 = (2o ·o 𝑥)))
2320, 22imbi12d 348 . . 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 6243 . . . . . . 7 (𝑦 = suc 𝑧 → suc 𝑦 = suc suc 𝑧)
2524eqeq1d 2826 . . . . . 6 (𝑦 = suc 𝑧 → (suc 𝑦 = (2o ·o 𝑥) ↔ suc suc 𝑧 = (2o ·o 𝑥)))
2625rexbidv 3289 . . . . 5 (𝑦 = suc 𝑧 → (∃𝑥 ∈ ω suc 𝑦 = (2o ·o 𝑥) ↔ ∃𝑥 ∈ ω suc suc 𝑧 = (2o ·o 𝑥)))
2726notbid 321 . . . 4 (𝑦 = suc 𝑧 → (¬ ∃𝑥 ∈ ω suc 𝑦 = (2o ·o 𝑥) ↔ ¬ ∃𝑥 ∈ ω suc suc 𝑧 = (2o ·o 𝑥)))
28 eqeq1 2828 . . . . 5 (𝑦 = suc 𝑧 → (𝑦 = (2o ·o 𝑥) ↔ suc 𝑧 = (2o ·o 𝑥)))
2928rexbidv 3289 . . . 4 (𝑦 = suc 𝑧 → (∃𝑥 ∈ ω 𝑦 = (2o ·o 𝑥) ↔ ∃𝑥 ∈ ω suc 𝑧 = (2o ·o 𝑥)))
3027, 29imbi12d 348 . . 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 6243 . . . . . . 7 (𝑦 = 𝐴 → suc 𝑦 = suc 𝐴)
3231eqeq1d 2826 . . . . . 6 (𝑦 = 𝐴 → (suc 𝑦 = (2o ·o 𝑥) ↔ suc 𝐴 = (2o ·o 𝑥)))
3332rexbidv 3289 . . . . 5 (𝑦 = 𝐴 → (∃𝑥 ∈ ω suc 𝑦 = (2o ·o 𝑥) ↔ ∃𝑥 ∈ ω suc 𝐴 = (2o ·o 𝑥)))
3433notbid 321 . . . 4 (𝑦 = 𝐴 → (¬ ∃𝑥 ∈ ω suc 𝑦 = (2o ·o 𝑥) ↔ ¬ ∃𝑥 ∈ ω suc 𝐴 = (2o ·o 𝑥)))
35 eqeq1 2828 . . . . 5 (𝑦 = 𝐴 → (𝑦 = (2o ·o 𝑥) ↔ 𝐴 = (2o ·o 𝑥)))
3635rexbidv 3289 . . . 4 (𝑦 = 𝐴 → (∃𝑥 ∈ ω 𝑦 = (2o ·o 𝑥) ↔ ∃𝑥 ∈ ω 𝐴 = (2o ·o 𝑥)))
3734, 36imbi12d 348 . . 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 7595 . . . . 5 ∅ ∈ ω
39 eqid 2824 . . . . 5 ∅ = ∅
40 oveq2 7157 . . . . . . 7 (𝑥 = ∅ → (2o ·o 𝑥) = (2o ·o ∅))
41 2on 8107 . . . . . . . 8 2o ∈ On
42 om0 8138 . . . . . . . 8 (2o ∈ On → (2o ·o ∅) = ∅)
4341, 42ax-mp 5 . . . . . . 7 (2o ·o ∅) = ∅
4440, 43syl6eq 2875 . . . . . 6 (𝑥 = ∅ → (2o ·o 𝑥) = ∅)
4544rspceeqv 3624 . . . . 5 ((∅ ∈ ω ∧ ∅ = ∅) → ∃𝑥 ∈ ω ∅ = (2o ·o 𝑥))
4638, 39, 45mp2an 691 . . . 4 𝑥 ∈ ω ∅ = (2o ·o 𝑥)
4746a1i 11 . . 3 (¬ ∃𝑥 ∈ ω suc ∅ = (2o ·o 𝑥) → ∃𝑥 ∈ ω ∅ = (2o ·o 𝑥))
481eqeq2d 2835 . . . . . . 7 (𝑥 = 𝑦 → (𝑧 = (2o ·o 𝑥) ↔ 𝑧 = (2o ·o 𝑦)))
4948cbvrexvw 3435 . . . . . 6 (∃𝑥 ∈ ω 𝑧 = (2o ·o 𝑥) ↔ ∃𝑦 ∈ ω 𝑧 = (2o ·o 𝑦))
50 peano2 7596 . . . . . . . . . 10 (𝑦 ∈ ω → suc 𝑦 ∈ ω)
51 2onn 8262 . . . . . . . . . . . 12 2o ∈ ω
52 nnmsuc 8229 . . . . . . . . . . . 12 ((2o ∈ ω ∧ 𝑦 ∈ ω) → (2o ·o suc 𝑦) = ((2o ·o 𝑦) +o 2o))
5351, 52mpan 689 . . . . . . . . . . 11 (𝑦 ∈ ω → (2o ·o suc 𝑦) = ((2o ·o 𝑦) +o 2o))
54 df-2o 8099 . . . . . . . . . . . . 13 2o = suc 1o
5554oveq2i 7160 . . . . . . . . . . . 12 ((2o ·o 𝑦) +o 2o) = ((2o ·o 𝑦) +o suc 1o)
56 nnmcl 8234 . . . . . . . . . . . . . 14 ((2o ∈ ω ∧ 𝑦 ∈ ω) → (2o ·o 𝑦) ∈ ω)
5751, 56mpan 689 . . . . . . . . . . . . 13 (𝑦 ∈ ω → (2o ·o 𝑦) ∈ ω)
58 1onn 8261 . . . . . . . . . . . . 13 1o ∈ ω
59 nnasuc 8228 . . . . . . . . . . . . 13 (((2o ·o 𝑦) ∈ ω ∧ 1o ∈ ω) → ((2o ·o 𝑦) +o suc 1o) = suc ((2o ·o 𝑦) +o 1o))
6057, 58, 59sylancl 589 . . . . . . . . . . . 12 (𝑦 ∈ ω → ((2o ·o 𝑦) +o suc 1o) = suc ((2o ·o 𝑦) +o 1o))
6155, 60syl5req 2872 . . . . . . . . . . 11 (𝑦 ∈ ω → suc ((2o ·o 𝑦) +o 1o) = ((2o ·o 𝑦) +o 2o))
62 nnon 7580 . . . . . . . . . . . 12 ((2o ·o 𝑦) ∈ ω → (2o ·o 𝑦) ∈ On)
63 oa1suc 8152 . . . . . . . . . . . 12 ((2o ·o 𝑦) ∈ On → ((2o ·o 𝑦) +o 1o) = suc (2o ·o 𝑦))
64 suceq 6243 . . . . . . . . . . . 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 2866 . . . . . . . . . 10 (𝑦 ∈ ω → suc suc (2o ·o 𝑦) = (2o ·o suc 𝑦))
67 oveq2 7157 . . . . . . . . . . 11 (𝑥 = suc 𝑦 → (2o ·o 𝑥) = (2o ·o suc 𝑦))
6867rspceeqv 3624 . . . . . . . . . 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 587 . . . . . . . . 9 (𝑦 ∈ ω → ∃𝑥 ∈ ω suc suc (2o ·o 𝑦) = (2o ·o 𝑥))
70 suceq 6243 . . . . . . . . . . . 12 (𝑧 = (2o ·o 𝑦) → suc 𝑧 = suc (2o ·o 𝑦))
71 suceq 6243 . . . . . . . . . . . 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 2826 . . . . . . . . . 10 (𝑧 = (2o ·o 𝑦) → (suc suc 𝑧 = (2o ·o 𝑥) ↔ suc suc (2o ·o 𝑦) = (2o ·o 𝑥)))
7473rexbidv 3289 . . . . . . . . 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 250 . . . . . . . 8 (𝑦 ∈ ω → (𝑧 = (2o ·o 𝑦) → ∃𝑥 ∈ ω suc suc 𝑧 = (2o ·o 𝑥)))
7675rexlimiv 3272 . . . . . . 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 245 . . . . 5 (𝑧 ∈ ω → (∃𝑥 ∈ ω 𝑧 = (2o ·o 𝑥) → ∃𝑥 ∈ ω suc suc 𝑧 = (2o ·o 𝑥)))
7978con3d 155 . . . 4 (𝑧 ∈ ω → (¬ ∃𝑥 ∈ ω suc suc 𝑧 = (2o ·o 𝑥) → ¬ ∃𝑥 ∈ ω 𝑧 = (2o ·o 𝑥)))
80 con1 148 . . . 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 7603 . 2 (𝐴 ∈ ω → (¬ ∃𝑥 ∈ ω suc 𝐴 = (2o ·o 𝑥) → ∃𝑥 ∈ ω 𝐴 = (2o ·o 𝑥)))
839, 82impbid2 229 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 209  wa 399   = wceq 1538  wcel 2115  wrex 3134  c0 4276  Oncon0 6178  suc csuc 6180  (class class class)co 7149  ωcom 7574  1oc1o 8091  2oc2o 8092   +o coa 8095   ·o comu 8096
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4825  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-tr 5159  df-id 5447  df-eprel 5452  df-po 5461  df-so 5462  df-fr 5501  df-we 5503  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-pred 6135  df-ord 6181  df-on 6182  df-lim 6183  df-suc 6184  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-ov 7152  df-oprab 7153  df-mpo 7154  df-om 7575  df-wrecs 7943  df-recs 8004  df-rdg 8042  df-1o 8098  df-2o 8099  df-oadd 8102  df-omul 8103
This theorem is referenced by:  fin1a2lem5  9824
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