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Theorem nneob 8695
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 7440 . . . . 5 (𝑥 = 𝑦 → (2o ·o 𝑥) = (2o ·o 𝑦))
21eqeq2d 2747 . . . 4 (𝑥 = 𝑦 → (𝐴 = (2o ·o 𝑥) ↔ 𝐴 = (2o ·o 𝑦)))
32cbvrexvw 3237 . . 3 (∃𝑥 ∈ ω 𝐴 = (2o ·o 𝑥) ↔ ∃𝑦 ∈ ω 𝐴 = (2o ·o 𝑦))
4 nnneo 8694 . . . . . . 7 ((𝑦 ∈ ω ∧ 𝑥 ∈ ω ∧ 𝐴 = (2o ·o 𝑦)) → ¬ suc 𝐴 = (2o ·o 𝑥))
543com23 1126 . . . . . 6 ((𝑦 ∈ ω ∧ 𝐴 = (2o ·o 𝑦) ∧ 𝑥 ∈ ω) → ¬ suc 𝐴 = (2o ·o 𝑥))
653expa 1118 . . . . 5 (((𝑦 ∈ ω ∧ 𝐴 = (2o ·o 𝑦)) ∧ 𝑥 ∈ ω) → ¬ suc 𝐴 = (2o ·o 𝑥))
76nrexdv 3148 . . . 4 ((𝑦 ∈ ω ∧ 𝐴 = (2o ·o 𝑦)) → ¬ ∃𝑥 ∈ ω suc 𝐴 = (2o ·o 𝑥))
87rexlimiva 3146 . . 3 (∃𝑦 ∈ ω 𝐴 = (2o ·o 𝑦) → ¬ ∃𝑥 ∈ ω suc 𝐴 = (2o ·o 𝑥))
93, 8sylbi 217 . 2 (∃𝑥 ∈ ω 𝐴 = (2o ·o 𝑥) → ¬ ∃𝑥 ∈ ω suc 𝐴 = (2o ·o 𝑥))
10 suceq 6449 . . . . . . 7 (𝑦 = ∅ → suc 𝑦 = suc ∅)
1110eqeq1d 2738 . . . . . 6 (𝑦 = ∅ → (suc 𝑦 = (2o ·o 𝑥) ↔ suc ∅ = (2o ·o 𝑥)))
1211rexbidv 3178 . . . . 5 (𝑦 = ∅ → (∃𝑥 ∈ ω suc 𝑦 = (2o ·o 𝑥) ↔ ∃𝑥 ∈ ω suc ∅ = (2o ·o 𝑥)))
1312notbid 318 . . . 4 (𝑦 = ∅ → (¬ ∃𝑥 ∈ ω suc 𝑦 = (2o ·o 𝑥) ↔ ¬ ∃𝑥 ∈ ω suc ∅ = (2o ·o 𝑥)))
14 eqeq1 2740 . . . . 5 (𝑦 = ∅ → (𝑦 = (2o ·o 𝑥) ↔ ∅ = (2o ·o 𝑥)))
1514rexbidv 3178 . . . 4 (𝑦 = ∅ → (∃𝑥 ∈ ω 𝑦 = (2o ·o 𝑥) ↔ ∃𝑥 ∈ ω ∅ = (2o ·o 𝑥)))
1613, 15imbi12d 344 . . 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 6449 . . . . . . 7 (𝑦 = 𝑧 → suc 𝑦 = suc 𝑧)
1817eqeq1d 2738 . . . . . 6 (𝑦 = 𝑧 → (suc 𝑦 = (2o ·o 𝑥) ↔ suc 𝑧 = (2o ·o 𝑥)))
1918rexbidv 3178 . . . . 5 (𝑦 = 𝑧 → (∃𝑥 ∈ ω suc 𝑦 = (2o ·o 𝑥) ↔ ∃𝑥 ∈ ω suc 𝑧 = (2o ·o 𝑥)))
2019notbid 318 . . . 4 (𝑦 = 𝑧 → (¬ ∃𝑥 ∈ ω suc 𝑦 = (2o ·o 𝑥) ↔ ¬ ∃𝑥 ∈ ω suc 𝑧 = (2o ·o 𝑥)))
21 eqeq1 2740 . . . . 5 (𝑦 = 𝑧 → (𝑦 = (2o ·o 𝑥) ↔ 𝑧 = (2o ·o 𝑥)))
2221rexbidv 3178 . . . 4 (𝑦 = 𝑧 → (∃𝑥 ∈ ω 𝑦 = (2o ·o 𝑥) ↔ ∃𝑥 ∈ ω 𝑧 = (2o ·o 𝑥)))
2320, 22imbi12d 344 . . 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 6449 . . . . . . 7 (𝑦 = suc 𝑧 → suc 𝑦 = suc suc 𝑧)
2524eqeq1d 2738 . . . . . 6 (𝑦 = suc 𝑧 → (suc 𝑦 = (2o ·o 𝑥) ↔ suc suc 𝑧 = (2o ·o 𝑥)))
2625rexbidv 3178 . . . . 5 (𝑦 = suc 𝑧 → (∃𝑥 ∈ ω suc 𝑦 = (2o ·o 𝑥) ↔ ∃𝑥 ∈ ω suc suc 𝑧 = (2o ·o 𝑥)))
2726notbid 318 . . . 4 (𝑦 = suc 𝑧 → (¬ ∃𝑥 ∈ ω suc 𝑦 = (2o ·o 𝑥) ↔ ¬ ∃𝑥 ∈ ω suc suc 𝑧 = (2o ·o 𝑥)))
28 eqeq1 2740 . . . . 5 (𝑦 = suc 𝑧 → (𝑦 = (2o ·o 𝑥) ↔ suc 𝑧 = (2o ·o 𝑥)))
2928rexbidv 3178 . . . 4 (𝑦 = suc 𝑧 → (∃𝑥 ∈ ω 𝑦 = (2o ·o 𝑥) ↔ ∃𝑥 ∈ ω suc 𝑧 = (2o ·o 𝑥)))
3027, 29imbi12d 344 . . 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 6449 . . . . . . 7 (𝑦 = 𝐴 → suc 𝑦 = suc 𝐴)
3231eqeq1d 2738 . . . . . 6 (𝑦 = 𝐴 → (suc 𝑦 = (2o ·o 𝑥) ↔ suc 𝐴 = (2o ·o 𝑥)))
3332rexbidv 3178 . . . . 5 (𝑦 = 𝐴 → (∃𝑥 ∈ ω suc 𝑦 = (2o ·o 𝑥) ↔ ∃𝑥 ∈ ω suc 𝐴 = (2o ·o 𝑥)))
3433notbid 318 . . . 4 (𝑦 = 𝐴 → (¬ ∃𝑥 ∈ ω suc 𝑦 = (2o ·o 𝑥) ↔ ¬ ∃𝑥 ∈ ω suc 𝐴 = (2o ·o 𝑥)))
35 eqeq1 2740 . . . . 5 (𝑦 = 𝐴 → (𝑦 = (2o ·o 𝑥) ↔ 𝐴 = (2o ·o 𝑥)))
3635rexbidv 3178 . . . 4 (𝑦 = 𝐴 → (∃𝑥 ∈ ω 𝑦 = (2o ·o 𝑥) ↔ ∃𝑥 ∈ ω 𝐴 = (2o ·o 𝑥)))
3734, 36imbi12d 344 . . 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 7911 . . . . 5 ∅ ∈ ω
39 eqid 2736 . . . . 5 ∅ = ∅
40 oveq2 7440 . . . . . . 7 (𝑥 = ∅ → (2o ·o 𝑥) = (2o ·o ∅))
41 2on 8521 . . . . . . . 8 2o ∈ On
42 om0 8556 . . . . . . . 8 (2o ∈ On → (2o ·o ∅) = ∅)
4341, 42ax-mp 5 . . . . . . 7 (2o ·o ∅) = ∅
4440, 43eqtrdi 2792 . . . . . 6 (𝑥 = ∅ → (2o ·o 𝑥) = ∅)
4544rspceeqv 3644 . . . . 5 ((∅ ∈ ω ∧ ∅ = ∅) → ∃𝑥 ∈ ω ∅ = (2o ·o 𝑥))
4638, 39, 45mp2an 692 . . . 4 𝑥 ∈ ω ∅ = (2o ·o 𝑥)
4746a1i 11 . . 3 (¬ ∃𝑥 ∈ ω suc ∅ = (2o ·o 𝑥) → ∃𝑥 ∈ ω ∅ = (2o ·o 𝑥))
481eqeq2d 2747 . . . . . . 7 (𝑥 = 𝑦 → (𝑧 = (2o ·o 𝑥) ↔ 𝑧 = (2o ·o 𝑦)))
4948cbvrexvw 3237 . . . . . 6 (∃𝑥 ∈ ω 𝑧 = (2o ·o 𝑥) ↔ ∃𝑦 ∈ ω 𝑧 = (2o ·o 𝑦))
50 peano2 7913 . . . . . . . . . 10 (𝑦 ∈ ω → suc 𝑦 ∈ ω)
51 2onn 8681 . . . . . . . . . . . 12 2o ∈ ω
52 nnmsuc 8646 . . . . . . . . . . . 12 ((2o ∈ ω ∧ 𝑦 ∈ ω) → (2o ·o suc 𝑦) = ((2o ·o 𝑦) +o 2o))
5351, 52mpan 690 . . . . . . . . . . 11 (𝑦 ∈ ω → (2o ·o suc 𝑦) = ((2o ·o 𝑦) +o 2o))
54 df-2o 8508 . . . . . . . . . . . . 13 2o = suc 1o
5554oveq2i 7443 . . . . . . . . . . . 12 ((2o ·o 𝑦) +o 2o) = ((2o ·o 𝑦) +o suc 1o)
56 nnmcl 8651 . . . . . . . . . . . . . 14 ((2o ∈ ω ∧ 𝑦 ∈ ω) → (2o ·o 𝑦) ∈ ω)
5751, 56mpan 690 . . . . . . . . . . . . 13 (𝑦 ∈ ω → (2o ·o 𝑦) ∈ ω)
58 1onn 8679 . . . . . . . . . . . . 13 1o ∈ ω
59 nnasuc 8645 . . . . . . . . . . . . 13 (((2o ·o 𝑦) ∈ ω ∧ 1o ∈ ω) → ((2o ·o 𝑦) +o suc 1o) = suc ((2o ·o 𝑦) +o 1o))
6057, 58, 59sylancl 586 . . . . . . . . . . . 12 (𝑦 ∈ ω → ((2o ·o 𝑦) +o suc 1o) = suc ((2o ·o 𝑦) +o 1o))
6155, 60eqtr2id 2789 . . . . . . . . . . 11 (𝑦 ∈ ω → suc ((2o ·o 𝑦) +o 1o) = ((2o ·o 𝑦) +o 2o))
62 nnon 7894 . . . . . . . . . . . 12 ((2o ·o 𝑦) ∈ ω → (2o ·o 𝑦) ∈ On)
63 oa1suc 8570 . . . . . . . . . . . 12 ((2o ·o 𝑦) ∈ On → ((2o ·o 𝑦) +o 1o) = suc (2o ·o 𝑦))
64 suceq 6449 . . . . . . . . . . . 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 2783 . . . . . . . . . 10 (𝑦 ∈ ω → suc suc (2o ·o 𝑦) = (2o ·o suc 𝑦))
67 oveq2 7440 . . . . . . . . . . 11 (𝑥 = suc 𝑦 → (2o ·o 𝑥) = (2o ·o suc 𝑦))
6867rspceeqv 3644 . . . . . . . . . 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 584 . . . . . . . . 9 (𝑦 ∈ ω → ∃𝑥 ∈ ω suc suc (2o ·o 𝑦) = (2o ·o 𝑥))
70 suceq 6449 . . . . . . . . . . . 12 (𝑧 = (2o ·o 𝑦) → suc 𝑧 = suc (2o ·o 𝑦))
71 suceq 6449 . . . . . . . . . . . 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 2738 . . . . . . . . . 10 (𝑧 = (2o ·o 𝑦) → (suc suc 𝑧 = (2o ·o 𝑥) ↔ suc suc (2o ·o 𝑦) = (2o ·o 𝑥)))
7473rexbidv 3178 . . . . . . . . 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 247 . . . . . . . 8 (𝑦 ∈ ω → (𝑧 = (2o ·o 𝑦) → ∃𝑥 ∈ ω suc suc 𝑧 = (2o ·o 𝑥)))
7675rexlimiv 3147 . . . . . . 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, 77biimtrid 242 . . . . 5 (𝑧 ∈ ω → (∃𝑥 ∈ ω 𝑧 = (2o ·o 𝑥) → ∃𝑥 ∈ ω suc suc 𝑧 = (2o ·o 𝑥)))
7978con3d 152 . . . 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 7919 . 2 (𝐴 ∈ ω → (¬ ∃𝑥 ∈ ω suc 𝐴 = (2o ·o 𝑥) → ∃𝑥 ∈ ω 𝐴 = (2o ·o 𝑥)))
839, 82impbid2 226 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 206  wa 395   = wceq 1539  wcel 2107  wrex 3069  c0 4332  Oncon0 6383  suc csuc 6385  (class class class)co 7432  ωcom 7888  1oc1o 8500  2oc2o 8501   +o coa 8504   ·o comu 8505
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-ov 7435  df-oprab 7436  df-mpo 7437  df-om 7889  df-2nd 8016  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-1o 8507  df-2o 8508  df-oadd 8511  df-omul 8512
This theorem is referenced by:  fin1a2lem5  10445
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