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Theorem nneob 8693
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 7439 . . . . 5 (𝑥 = 𝑦 → (2o ·o 𝑥) = (2o ·o 𝑦))
21eqeq2d 2746 . . . 4 (𝑥 = 𝑦 → (𝐴 = (2o ·o 𝑥) ↔ 𝐴 = (2o ·o 𝑦)))
32cbvrexvw 3236 . . 3 (∃𝑥 ∈ ω 𝐴 = (2o ·o 𝑥) ↔ ∃𝑦 ∈ ω 𝐴 = (2o ·o 𝑦))
4 nnneo 8692 . . . . . . 7 ((𝑦 ∈ ω ∧ 𝑥 ∈ ω ∧ 𝐴 = (2o ·o 𝑦)) → ¬ suc 𝐴 = (2o ·o 𝑥))
543com23 1125 . . . . . 6 ((𝑦 ∈ ω ∧ 𝐴 = (2o ·o 𝑦) ∧ 𝑥 ∈ ω) → ¬ suc 𝐴 = (2o ·o 𝑥))
653expa 1117 . . . . 5 (((𝑦 ∈ ω ∧ 𝐴 = (2o ·o 𝑦)) ∧ 𝑥 ∈ ω) → ¬ suc 𝐴 = (2o ·o 𝑥))
76nrexdv 3147 . . . 4 ((𝑦 ∈ ω ∧ 𝐴 = (2o ·o 𝑦)) → ¬ ∃𝑥 ∈ ω suc 𝐴 = (2o ·o 𝑥))
87rexlimiva 3145 . . 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 6452 . . . . . . 7 (𝑦 = ∅ → suc 𝑦 = suc ∅)
1110eqeq1d 2737 . . . . . 6 (𝑦 = ∅ → (suc 𝑦 = (2o ·o 𝑥) ↔ suc ∅ = (2o ·o 𝑥)))
1211rexbidv 3177 . . . . 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 2739 . . . . 5 (𝑦 = ∅ → (𝑦 = (2o ·o 𝑥) ↔ ∅ = (2o ·o 𝑥)))
1514rexbidv 3177 . . . 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 6452 . . . . . . 7 (𝑦 = 𝑧 → suc 𝑦 = suc 𝑧)
1817eqeq1d 2737 . . . . . 6 (𝑦 = 𝑧 → (suc 𝑦 = (2o ·o 𝑥) ↔ suc 𝑧 = (2o ·o 𝑥)))
1918rexbidv 3177 . . . . 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 2739 . . . . 5 (𝑦 = 𝑧 → (𝑦 = (2o ·o 𝑥) ↔ 𝑧 = (2o ·o 𝑥)))
2221rexbidv 3177 . . . 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 6452 . . . . . . 7 (𝑦 = suc 𝑧 → suc 𝑦 = suc suc 𝑧)
2524eqeq1d 2737 . . . . . 6 (𝑦 = suc 𝑧 → (suc 𝑦 = (2o ·o 𝑥) ↔ suc suc 𝑧 = (2o ·o 𝑥)))
2625rexbidv 3177 . . . . 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 2739 . . . . 5 (𝑦 = suc 𝑧 → (𝑦 = (2o ·o 𝑥) ↔ suc 𝑧 = (2o ·o 𝑥)))
2928rexbidv 3177 . . . 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 6452 . . . . . . 7 (𝑦 = 𝐴 → suc 𝑦 = suc 𝐴)
3231eqeq1d 2737 . . . . . 6 (𝑦 = 𝐴 → (suc 𝑦 = (2o ·o 𝑥) ↔ suc 𝐴 = (2o ·o 𝑥)))
3332rexbidv 3177 . . . . 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 2739 . . . . 5 (𝑦 = 𝐴 → (𝑦 = (2o ·o 𝑥) ↔ 𝐴 = (2o ·o 𝑥)))
3635rexbidv 3177 . . . 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 2735 . . . . 5 ∅ = ∅
40 oveq2 7439 . . . . . . 7 (𝑥 = ∅ → (2o ·o 𝑥) = (2o ·o ∅))
41 2on 8519 . . . . . . . 8 2o ∈ On
42 om0 8554 . . . . . . . 8 (2o ∈ On → (2o ·o ∅) = ∅)
4341, 42ax-mp 5 . . . . . . 7 (2o ·o ∅) = ∅
4440, 43eqtrdi 2791 . . . . . 6 (𝑥 = ∅ → (2o ·o 𝑥) = ∅)
4544rspceeqv 3645 . . . . 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 2746 . . . . . . 7 (𝑥 = 𝑦 → (𝑧 = (2o ·o 𝑥) ↔ 𝑧 = (2o ·o 𝑦)))
4948cbvrexvw 3236 . . . . . 6 (∃𝑥 ∈ ω 𝑧 = (2o ·o 𝑥) ↔ ∃𝑦 ∈ ω 𝑧 = (2o ·o 𝑦))
50 peano2 7913 . . . . . . . . . 10 (𝑦 ∈ ω → suc 𝑦 ∈ ω)
51 2onn 8679 . . . . . . . . . . . 12 2o ∈ ω
52 nnmsuc 8644 . . . . . . . . . . . 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 8506 . . . . . . . . . . . . 13 2o = suc 1o
5554oveq2i 7442 . . . . . . . . . . . 12 ((2o ·o 𝑦) +o 2o) = ((2o ·o 𝑦) +o suc 1o)
56 nnmcl 8649 . . . . . . . . . . . . . 14 ((2o ∈ ω ∧ 𝑦 ∈ ω) → (2o ·o 𝑦) ∈ ω)
5751, 56mpan 690 . . . . . . . . . . . . 13 (𝑦 ∈ ω → (2o ·o 𝑦) ∈ ω)
58 1onn 8677 . . . . . . . . . . . . 13 1o ∈ ω
59 nnasuc 8643 . . . . . . . . . . . . 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 2788 . . . . . . . . . . 11 (𝑦 ∈ ω → suc ((2o ·o 𝑦) +o 1o) = ((2o ·o 𝑦) +o 2o))
62 nnon 7893 . . . . . . . . . . . 12 ((2o ·o 𝑦) ∈ ω → (2o ·o 𝑦) ∈ On)
63 oa1suc 8568 . . . . . . . . . . . 12 ((2o ·o 𝑦) ∈ On → ((2o ·o 𝑦) +o 1o) = suc (2o ·o 𝑦))
64 suceq 6452 . . . . . . . . . . . 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 2782 . . . . . . . . . 10 (𝑦 ∈ ω → suc suc (2o ·o 𝑦) = (2o ·o suc 𝑦))
67 oveq2 7439 . . . . . . . . . . 11 (𝑥 = suc 𝑦 → (2o ·o 𝑥) = (2o ·o suc 𝑦))
6867rspceeqv 3645 . . . . . . . . . 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 6452 . . . . . . . . . . . 12 (𝑧 = (2o ·o 𝑦) → suc 𝑧 = suc (2o ·o 𝑦))
71 suceq 6452 . . . . . . . . . . . 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 2737 . . . . . . . . . 10 (𝑧 = (2o ·o 𝑦) → (suc suc 𝑧 = (2o ·o 𝑥) ↔ suc suc (2o ·o 𝑦) = (2o ·o 𝑥)))
7473rexbidv 3177 . . . . . . . . 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 3146 . . . . . . 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 1537  wcel 2106  wrex 3068  c0 4339  Oncon0 6386  suc csuc 6388  (class class class)co 7431  ωcom 7887  1oc1o 8498  2oc2o 8499   +o coa 8502   ·o comu 8503
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-2o 8506  df-oadd 8509  df-omul 8510
This theorem is referenced by:  fin1a2lem5  10442
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