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Theorem nneob 7886
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 6801 . . . . 5 (𝑥 = 𝑦 → (2𝑜 ·𝑜 𝑥) = (2𝑜 ·𝑜 𝑦))
21eqeq2d 2781 . . . 4 (𝑥 = 𝑦 → (𝐴 = (2𝑜 ·𝑜 𝑥) ↔ 𝐴 = (2𝑜 ·𝑜 𝑦)))
32cbvrexv 3321 . . 3 (∃𝑥 ∈ ω 𝐴 = (2𝑜 ·𝑜 𝑥) ↔ ∃𝑦 ∈ ω 𝐴 = (2𝑜 ·𝑜 𝑦))
4 nnneo 7885 . . . . . . 7 ((𝑦 ∈ ω ∧ 𝑥 ∈ ω ∧ 𝐴 = (2𝑜 ·𝑜 𝑦)) → ¬ suc 𝐴 = (2𝑜 ·𝑜 𝑥))
543com23 1120 . . . . . 6 ((𝑦 ∈ ω ∧ 𝐴 = (2𝑜 ·𝑜 𝑦) ∧ 𝑥 ∈ ω) → ¬ suc 𝐴 = (2𝑜 ·𝑜 𝑥))
653expa 1111 . . . . 5 (((𝑦 ∈ ω ∧ 𝐴 = (2𝑜 ·𝑜 𝑦)) ∧ 𝑥 ∈ ω) → ¬ suc 𝐴 = (2𝑜 ·𝑜 𝑥))
76nrexdv 3149 . . . 4 ((𝑦 ∈ ω ∧ 𝐴 = (2𝑜 ·𝑜 𝑦)) → ¬ ∃𝑥 ∈ ω suc 𝐴 = (2𝑜 ·𝑜 𝑥))
87rexlimiva 3176 . . 3 (∃𝑦 ∈ ω 𝐴 = (2𝑜 ·𝑜 𝑦) → ¬ ∃𝑥 ∈ ω suc 𝐴 = (2𝑜 ·𝑜 𝑥))
93, 8sylbi 207 . 2 (∃𝑥 ∈ ω 𝐴 = (2𝑜 ·𝑜 𝑥) → ¬ ∃𝑥 ∈ ω suc 𝐴 = (2𝑜 ·𝑜 𝑥))
10 suceq 5933 . . . . . . 7 (𝑦 = ∅ → suc 𝑦 = suc ∅)
1110eqeq1d 2773 . . . . . 6 (𝑦 = ∅ → (suc 𝑦 = (2𝑜 ·𝑜 𝑥) ↔ suc ∅ = (2𝑜 ·𝑜 𝑥)))
1211rexbidv 3200 . . . . 5 (𝑦 = ∅ → (∃𝑥 ∈ ω suc 𝑦 = (2𝑜 ·𝑜 𝑥) ↔ ∃𝑥 ∈ ω suc ∅ = (2𝑜 ·𝑜 𝑥)))
1312notbid 307 . . . 4 (𝑦 = ∅ → (¬ ∃𝑥 ∈ ω suc 𝑦 = (2𝑜 ·𝑜 𝑥) ↔ ¬ ∃𝑥 ∈ ω suc ∅ = (2𝑜 ·𝑜 𝑥)))
14 eqeq1 2775 . . . . 5 (𝑦 = ∅ → (𝑦 = (2𝑜 ·𝑜 𝑥) ↔ ∅ = (2𝑜 ·𝑜 𝑥)))
1514rexbidv 3200 . . . 4 (𝑦 = ∅ → (∃𝑥 ∈ ω 𝑦 = (2𝑜 ·𝑜 𝑥) ↔ ∃𝑥 ∈ ω ∅ = (2𝑜 ·𝑜 𝑥)))
1613, 15imbi12d 333 . . 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 5933 . . . . . . 7 (𝑦 = 𝑧 → suc 𝑦 = suc 𝑧)
1817eqeq1d 2773 . . . . . 6 (𝑦 = 𝑧 → (suc 𝑦 = (2𝑜 ·𝑜 𝑥) ↔ suc 𝑧 = (2𝑜 ·𝑜 𝑥)))
1918rexbidv 3200 . . . . 5 (𝑦 = 𝑧 → (∃𝑥 ∈ ω suc 𝑦 = (2𝑜 ·𝑜 𝑥) ↔ ∃𝑥 ∈ ω suc 𝑧 = (2𝑜 ·𝑜 𝑥)))
2019notbid 307 . . . 4 (𝑦 = 𝑧 → (¬ ∃𝑥 ∈ ω suc 𝑦 = (2𝑜 ·𝑜 𝑥) ↔ ¬ ∃𝑥 ∈ ω suc 𝑧 = (2𝑜 ·𝑜 𝑥)))
21 eqeq1 2775 . . . . 5 (𝑦 = 𝑧 → (𝑦 = (2𝑜 ·𝑜 𝑥) ↔ 𝑧 = (2𝑜 ·𝑜 𝑥)))
2221rexbidv 3200 . . . 4 (𝑦 = 𝑧 → (∃𝑥 ∈ ω 𝑦 = (2𝑜 ·𝑜 𝑥) ↔ ∃𝑥 ∈ ω 𝑧 = (2𝑜 ·𝑜 𝑥)))
2320, 22imbi12d 333 . . 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 5933 . . . . . . 7 (𝑦 = suc 𝑧 → suc 𝑦 = suc suc 𝑧)
2524eqeq1d 2773 . . . . . 6 (𝑦 = suc 𝑧 → (suc 𝑦 = (2𝑜 ·𝑜 𝑥) ↔ suc suc 𝑧 = (2𝑜 ·𝑜 𝑥)))
2625rexbidv 3200 . . . . 5 (𝑦 = suc 𝑧 → (∃𝑥 ∈ ω suc 𝑦 = (2𝑜 ·𝑜 𝑥) ↔ ∃𝑥 ∈ ω suc suc 𝑧 = (2𝑜 ·𝑜 𝑥)))
2726notbid 307 . . . 4 (𝑦 = suc 𝑧 → (¬ ∃𝑥 ∈ ω suc 𝑦 = (2𝑜 ·𝑜 𝑥) ↔ ¬ ∃𝑥 ∈ ω suc suc 𝑧 = (2𝑜 ·𝑜 𝑥)))
28 eqeq1 2775 . . . . 5 (𝑦 = suc 𝑧 → (𝑦 = (2𝑜 ·𝑜 𝑥) ↔ suc 𝑧 = (2𝑜 ·𝑜 𝑥)))
2928rexbidv 3200 . . . 4 (𝑦 = suc 𝑧 → (∃𝑥 ∈ ω 𝑦 = (2𝑜 ·𝑜 𝑥) ↔ ∃𝑥 ∈ ω suc 𝑧 = (2𝑜 ·𝑜 𝑥)))
3027, 29imbi12d 333 . . 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 5933 . . . . . . 7 (𝑦 = 𝐴 → suc 𝑦 = suc 𝐴)
3231eqeq1d 2773 . . . . . 6 (𝑦 = 𝐴 → (suc 𝑦 = (2𝑜 ·𝑜 𝑥) ↔ suc 𝐴 = (2𝑜 ·𝑜 𝑥)))
3332rexbidv 3200 . . . . 5 (𝑦 = 𝐴 → (∃𝑥 ∈ ω suc 𝑦 = (2𝑜 ·𝑜 𝑥) ↔ ∃𝑥 ∈ ω suc 𝐴 = (2𝑜 ·𝑜 𝑥)))
3433notbid 307 . . . 4 (𝑦 = 𝐴 → (¬ ∃𝑥 ∈ ω suc 𝑦 = (2𝑜 ·𝑜 𝑥) ↔ ¬ ∃𝑥 ∈ ω suc 𝐴 = (2𝑜 ·𝑜 𝑥)))
35 eqeq1 2775 . . . . 5 (𝑦 = 𝐴 → (𝑦 = (2𝑜 ·𝑜 𝑥) ↔ 𝐴 = (2𝑜 ·𝑜 𝑥)))
3635rexbidv 3200 . . . 4 (𝑦 = 𝐴 → (∃𝑥 ∈ ω 𝑦 = (2𝑜 ·𝑜 𝑥) ↔ ∃𝑥 ∈ ω 𝐴 = (2𝑜 ·𝑜 𝑥)))
3734, 36imbi12d 333 . . 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 7232 . . . . 5 ∅ ∈ ω
39 eqid 2771 . . . . 5 ∅ = ∅
40 oveq2 6801 . . . . . . . 8 (𝑥 = ∅ → (2𝑜 ·𝑜 𝑥) = (2𝑜 ·𝑜 ∅))
41 om0x 7753 . . . . . . . 8 (2𝑜 ·𝑜 ∅) = ∅
4240, 41syl6eq 2821 . . . . . . 7 (𝑥 = ∅ → (2𝑜 ·𝑜 𝑥) = ∅)
4342eqeq2d 2781 . . . . . 6 (𝑥 = ∅ → (∅ = (2𝑜 ·𝑜 𝑥) ↔ ∅ = ∅))
4443rspcev 3460 . . . . 5 ((∅ ∈ ω ∧ ∅ = ∅) → ∃𝑥 ∈ ω ∅ = (2𝑜 ·𝑜 𝑥))
4538, 39, 44mp2an 672 . . . 4 𝑥 ∈ ω ∅ = (2𝑜 ·𝑜 𝑥)
4645a1i 11 . . 3 (¬ ∃𝑥 ∈ ω suc ∅ = (2𝑜 ·𝑜 𝑥) → ∃𝑥 ∈ ω ∅ = (2𝑜 ·𝑜 𝑥))
471eqeq2d 2781 . . . . . . 7 (𝑥 = 𝑦 → (𝑧 = (2𝑜 ·𝑜 𝑥) ↔ 𝑧 = (2𝑜 ·𝑜 𝑦)))
4847cbvrexv 3321 . . . . . 6 (∃𝑥 ∈ ω 𝑧 = (2𝑜 ·𝑜 𝑥) ↔ ∃𝑦 ∈ ω 𝑧 = (2𝑜 ·𝑜 𝑦))
49 peano2 7233 . . . . . . . . . 10 (𝑦 ∈ ω → suc 𝑦 ∈ ω)
50 2onn 7874 . . . . . . . . . . . 12 2𝑜 ∈ ω
51 nnmsuc 7841 . . . . . . . . . . . 12 ((2𝑜 ∈ ω ∧ 𝑦 ∈ ω) → (2𝑜 ·𝑜 suc 𝑦) = ((2𝑜 ·𝑜 𝑦) +𝑜 2𝑜))
5250, 51mpan 670 . . . . . . . . . . 11 (𝑦 ∈ ω → (2𝑜 ·𝑜 suc 𝑦) = ((2𝑜 ·𝑜 𝑦) +𝑜 2𝑜))
53 df-2o 7714 . . . . . . . . . . . . 13 2𝑜 = suc 1𝑜
5453oveq2i 6804 . . . . . . . . . . . 12 ((2𝑜 ·𝑜 𝑦) +𝑜 2𝑜) = ((2𝑜 ·𝑜 𝑦) +𝑜 suc 1𝑜)
55 nnmcl 7846 . . . . . . . . . . . . . 14 ((2𝑜 ∈ ω ∧ 𝑦 ∈ ω) → (2𝑜 ·𝑜 𝑦) ∈ ω)
5650, 55mpan 670 . . . . . . . . . . . . 13 (𝑦 ∈ ω → (2𝑜 ·𝑜 𝑦) ∈ ω)
57 1onn 7873 . . . . . . . . . . . . 13 1𝑜 ∈ ω
58 nnasuc 7840 . . . . . . . . . . . . 13 (((2𝑜 ·𝑜 𝑦) ∈ ω ∧ 1𝑜 ∈ ω) → ((2𝑜 ·𝑜 𝑦) +𝑜 suc 1𝑜) = suc ((2𝑜 ·𝑜 𝑦) +𝑜 1𝑜))
5956, 57, 58sylancl 574 . . . . . . . . . . . 12 (𝑦 ∈ ω → ((2𝑜 ·𝑜 𝑦) +𝑜 suc 1𝑜) = suc ((2𝑜 ·𝑜 𝑦) +𝑜 1𝑜))
6054, 59syl5req 2818 . . . . . . . . . . 11 (𝑦 ∈ ω → suc ((2𝑜 ·𝑜 𝑦) +𝑜 1𝑜) = ((2𝑜 ·𝑜 𝑦) +𝑜 2𝑜))
61 nnon 7218 . . . . . . . . . . . 12 ((2𝑜 ·𝑜 𝑦) ∈ ω → (2𝑜 ·𝑜 𝑦) ∈ On)
62 oa1suc 7765 . . . . . . . . . . . 12 ((2𝑜 ·𝑜 𝑦) ∈ On → ((2𝑜 ·𝑜 𝑦) +𝑜 1𝑜) = suc (2𝑜 ·𝑜 𝑦))
63 suceq 5933 . . . . . . . . . . . 12 (((2𝑜 ·𝑜 𝑦) +𝑜 1𝑜) = suc (2𝑜 ·𝑜 𝑦) → suc ((2𝑜 ·𝑜 𝑦) +𝑜 1𝑜) = suc suc (2𝑜 ·𝑜 𝑦))
6456, 61, 62, 634syl 19 . . . . . . . . . . 11 (𝑦 ∈ ω → suc ((2𝑜 ·𝑜 𝑦) +𝑜 1𝑜) = suc suc (2𝑜 ·𝑜 𝑦))
6552, 60, 643eqtr2rd 2812 . . . . . . . . . 10 (𝑦 ∈ ω → suc suc (2𝑜 ·𝑜 𝑦) = (2𝑜 ·𝑜 suc 𝑦))
66 oveq2 6801 . . . . . . . . . . . 12 (𝑥 = suc 𝑦 → (2𝑜 ·𝑜 𝑥) = (2𝑜 ·𝑜 suc 𝑦))
6766eqeq2d 2781 . . . . . . . . . . 11 (𝑥 = suc 𝑦 → (suc suc (2𝑜 ·𝑜 𝑦) = (2𝑜 ·𝑜 𝑥) ↔ suc suc (2𝑜 ·𝑜 𝑦) = (2𝑜 ·𝑜 suc 𝑦)))
6867rspcev 3460 . . . . . . . . . 10 ((suc 𝑦 ∈ ω ∧ suc suc (2𝑜 ·𝑜 𝑦) = (2𝑜 ·𝑜 suc 𝑦)) → ∃𝑥 ∈ ω suc suc (2𝑜 ·𝑜 𝑦) = (2𝑜 ·𝑜 𝑥))
6949, 65, 68syl2anc 573 . . . . . . . . 9 (𝑦 ∈ ω → ∃𝑥 ∈ ω suc suc (2𝑜 ·𝑜 𝑦) = (2𝑜 ·𝑜 𝑥))
70 suceq 5933 . . . . . . . . . . . 12 (𝑧 = (2𝑜 ·𝑜 𝑦) → suc 𝑧 = suc (2𝑜 ·𝑜 𝑦))
71 suceq 5933 . . . . . . . . . . . 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 2773 . . . . . . . . . 10 (𝑧 = (2𝑜 ·𝑜 𝑦) → (suc suc 𝑧 = (2𝑜 ·𝑜 𝑥) ↔ suc suc (2𝑜 ·𝑜 𝑦) = (2𝑜 ·𝑜 𝑥)))
7473rexbidv 3200 . . . . . . . . 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 237 . . . . . . . 8 (𝑦 ∈ ω → (𝑧 = (2𝑜 ·𝑜 𝑦) → ∃𝑥 ∈ ω suc suc 𝑧 = (2𝑜 ·𝑜 𝑥)))
7675rexlimiv 3175 . . . . . . 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)))
7848, 77syl5bi 232 . . . . 5 (𝑧 ∈ ω → (∃𝑥 ∈ ω 𝑧 = (2𝑜 ·𝑜 𝑥) → ∃𝑥 ∈ ω suc suc 𝑧 = (2𝑜 ·𝑜 𝑥)))
7978con3d 149 . . . 4 (𝑧 ∈ ω → (¬ ∃𝑥 ∈ ω suc suc 𝑧 = (2𝑜 ·𝑜 𝑥) → ¬ ∃𝑥 ∈ ω 𝑧 = (2𝑜 ·𝑜 𝑥)))
80 con1 145 . . . 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, 46, 81finds 7239 . 2 (𝐴 ∈ ω → (¬ ∃𝑥 ∈ ω suc 𝐴 = (2𝑜 ·𝑜 𝑥) → ∃𝑥 ∈ ω 𝐴 = (2𝑜 ·𝑜 𝑥)))
839, 82impbid2 216 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 196  wa 382   = wceq 1631  wcel 2145  wrex 3062  c0 4063  Oncon0 5866  suc csuc 5868  (class class class)co 6793  ωcom 7212  1𝑜c1o 7706  2𝑜c2o 7707   +𝑜 coa 7710   ·𝑜 comu 7711
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-om 7213  df-1st 7315  df-2nd 7316  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-1o 7713  df-2o 7714  df-oadd 7717  df-omul 7718
This theorem is referenced by:  fin1a2lem5  9428
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