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Theorem nneob 7677
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 6612 . . . . 5 (𝑥 = 𝑦 → (2𝑜 ·𝑜 𝑥) = (2𝑜 ·𝑜 𝑦))
21eqeq2d 2631 . . . 4 (𝑥 = 𝑦 → (𝐴 = (2𝑜 ·𝑜 𝑥) ↔ 𝐴 = (2𝑜 ·𝑜 𝑦)))
32cbvrexv 3160 . . 3 (∃𝑥 ∈ ω 𝐴 = (2𝑜 ·𝑜 𝑥) ↔ ∃𝑦 ∈ ω 𝐴 = (2𝑜 ·𝑜 𝑦))
4 nnneo 7676 . . . . . . 7 ((𝑦 ∈ ω ∧ 𝑥 ∈ ω ∧ 𝐴 = (2𝑜 ·𝑜 𝑦)) → ¬ suc 𝐴 = (2𝑜 ·𝑜 𝑥))
543com23 1268 . . . . . 6 ((𝑦 ∈ ω ∧ 𝐴 = (2𝑜 ·𝑜 𝑦) ∧ 𝑥 ∈ ω) → ¬ suc 𝐴 = (2𝑜 ·𝑜 𝑥))
653expa 1262 . . . . 5 (((𝑦 ∈ ω ∧ 𝐴 = (2𝑜 ·𝑜 𝑦)) ∧ 𝑥 ∈ ω) → ¬ suc 𝐴 = (2𝑜 ·𝑜 𝑥))
76nrexdv 2995 . . . 4 ((𝑦 ∈ ω ∧ 𝐴 = (2𝑜 ·𝑜 𝑦)) → ¬ ∃𝑥 ∈ ω suc 𝐴 = (2𝑜 ·𝑜 𝑥))
87rexlimiva 3021 . . 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 5749 . . . . . . 7 (𝑦 = ∅ → suc 𝑦 = suc ∅)
1110eqeq1d 2623 . . . . . 6 (𝑦 = ∅ → (suc 𝑦 = (2𝑜 ·𝑜 𝑥) ↔ suc ∅ = (2𝑜 ·𝑜 𝑥)))
1211rexbidv 3045 . . . . 5 (𝑦 = ∅ → (∃𝑥 ∈ ω suc 𝑦 = (2𝑜 ·𝑜 𝑥) ↔ ∃𝑥 ∈ ω suc ∅ = (2𝑜 ·𝑜 𝑥)))
1312notbid 308 . . . 4 (𝑦 = ∅ → (¬ ∃𝑥 ∈ ω suc 𝑦 = (2𝑜 ·𝑜 𝑥) ↔ ¬ ∃𝑥 ∈ ω suc ∅ = (2𝑜 ·𝑜 𝑥)))
14 eqeq1 2625 . . . . 5 (𝑦 = ∅ → (𝑦 = (2𝑜 ·𝑜 𝑥) ↔ ∅ = (2𝑜 ·𝑜 𝑥)))
1514rexbidv 3045 . . . 4 (𝑦 = ∅ → (∃𝑥 ∈ ω 𝑦 = (2𝑜 ·𝑜 𝑥) ↔ ∃𝑥 ∈ ω ∅ = (2𝑜 ·𝑜 𝑥)))
1613, 15imbi12d 334 . . 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 5749 . . . . . . 7 (𝑦 = 𝑧 → suc 𝑦 = suc 𝑧)
1817eqeq1d 2623 . . . . . 6 (𝑦 = 𝑧 → (suc 𝑦 = (2𝑜 ·𝑜 𝑥) ↔ suc 𝑧 = (2𝑜 ·𝑜 𝑥)))
1918rexbidv 3045 . . . . 5 (𝑦 = 𝑧 → (∃𝑥 ∈ ω suc 𝑦 = (2𝑜 ·𝑜 𝑥) ↔ ∃𝑥 ∈ ω suc 𝑧 = (2𝑜 ·𝑜 𝑥)))
2019notbid 308 . . . 4 (𝑦 = 𝑧 → (¬ ∃𝑥 ∈ ω suc 𝑦 = (2𝑜 ·𝑜 𝑥) ↔ ¬ ∃𝑥 ∈ ω suc 𝑧 = (2𝑜 ·𝑜 𝑥)))
21 eqeq1 2625 . . . . 5 (𝑦 = 𝑧 → (𝑦 = (2𝑜 ·𝑜 𝑥) ↔ 𝑧 = (2𝑜 ·𝑜 𝑥)))
2221rexbidv 3045 . . . 4 (𝑦 = 𝑧 → (∃𝑥 ∈ ω 𝑦 = (2𝑜 ·𝑜 𝑥) ↔ ∃𝑥 ∈ ω 𝑧 = (2𝑜 ·𝑜 𝑥)))
2320, 22imbi12d 334 . . 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 5749 . . . . . . 7 (𝑦 = suc 𝑧 → suc 𝑦 = suc suc 𝑧)
2524eqeq1d 2623 . . . . . 6 (𝑦 = suc 𝑧 → (suc 𝑦 = (2𝑜 ·𝑜 𝑥) ↔ suc suc 𝑧 = (2𝑜 ·𝑜 𝑥)))
2625rexbidv 3045 . . . . 5 (𝑦 = suc 𝑧 → (∃𝑥 ∈ ω suc 𝑦 = (2𝑜 ·𝑜 𝑥) ↔ ∃𝑥 ∈ ω suc suc 𝑧 = (2𝑜 ·𝑜 𝑥)))
2726notbid 308 . . . 4 (𝑦 = suc 𝑧 → (¬ ∃𝑥 ∈ ω suc 𝑦 = (2𝑜 ·𝑜 𝑥) ↔ ¬ ∃𝑥 ∈ ω suc suc 𝑧 = (2𝑜 ·𝑜 𝑥)))
28 eqeq1 2625 . . . . 5 (𝑦 = suc 𝑧 → (𝑦 = (2𝑜 ·𝑜 𝑥) ↔ suc 𝑧 = (2𝑜 ·𝑜 𝑥)))
2928rexbidv 3045 . . . 4 (𝑦 = suc 𝑧 → (∃𝑥 ∈ ω 𝑦 = (2𝑜 ·𝑜 𝑥) ↔ ∃𝑥 ∈ ω suc 𝑧 = (2𝑜 ·𝑜 𝑥)))
3027, 29imbi12d 334 . . 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 5749 . . . . . . 7 (𝑦 = 𝐴 → suc 𝑦 = suc 𝐴)
3231eqeq1d 2623 . . . . . 6 (𝑦 = 𝐴 → (suc 𝑦 = (2𝑜 ·𝑜 𝑥) ↔ suc 𝐴 = (2𝑜 ·𝑜 𝑥)))
3332rexbidv 3045 . . . . 5 (𝑦 = 𝐴 → (∃𝑥 ∈ ω suc 𝑦 = (2𝑜 ·𝑜 𝑥) ↔ ∃𝑥 ∈ ω suc 𝐴 = (2𝑜 ·𝑜 𝑥)))
3433notbid 308 . . . 4 (𝑦 = 𝐴 → (¬ ∃𝑥 ∈ ω suc 𝑦 = (2𝑜 ·𝑜 𝑥) ↔ ¬ ∃𝑥 ∈ ω suc 𝐴 = (2𝑜 ·𝑜 𝑥)))
35 eqeq1 2625 . . . . 5 (𝑦 = 𝐴 → (𝑦 = (2𝑜 ·𝑜 𝑥) ↔ 𝐴 = (2𝑜 ·𝑜 𝑥)))
3635rexbidv 3045 . . . 4 (𝑦 = 𝐴 → (∃𝑥 ∈ ω 𝑦 = (2𝑜 ·𝑜 𝑥) ↔ ∃𝑥 ∈ ω 𝐴 = (2𝑜 ·𝑜 𝑥)))
3734, 36imbi12d 334 . . 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 7032 . . . . 5 ∅ ∈ ω
39 eqid 2621 . . . . 5 ∅ = ∅
40 oveq2 6612 . . . . . . . 8 (𝑥 = ∅ → (2𝑜 ·𝑜 𝑥) = (2𝑜 ·𝑜 ∅))
41 om0x 7544 . . . . . . . 8 (2𝑜 ·𝑜 ∅) = ∅
4240, 41syl6eq 2671 . . . . . . 7 (𝑥 = ∅ → (2𝑜 ·𝑜 𝑥) = ∅)
4342eqeq2d 2631 . . . . . 6 (𝑥 = ∅ → (∅ = (2𝑜 ·𝑜 𝑥) ↔ ∅ = ∅))
4443rspcev 3295 . . . . 5 ((∅ ∈ ω ∧ ∅ = ∅) → ∃𝑥 ∈ ω ∅ = (2𝑜 ·𝑜 𝑥))
4538, 39, 44mp2an 707 . . . 4 𝑥 ∈ ω ∅ = (2𝑜 ·𝑜 𝑥)
4645a1i 11 . . 3 (¬ ∃𝑥 ∈ ω suc ∅ = (2𝑜 ·𝑜 𝑥) → ∃𝑥 ∈ ω ∅ = (2𝑜 ·𝑜 𝑥))
471eqeq2d 2631 . . . . . . 7 (𝑥 = 𝑦 → (𝑧 = (2𝑜 ·𝑜 𝑥) ↔ 𝑧 = (2𝑜 ·𝑜 𝑦)))
4847cbvrexv 3160 . . . . . 6 (∃𝑥 ∈ ω 𝑧 = (2𝑜 ·𝑜 𝑥) ↔ ∃𝑦 ∈ ω 𝑧 = (2𝑜 ·𝑜 𝑦))
49 peano2 7033 . . . . . . . . . 10 (𝑦 ∈ ω → suc 𝑦 ∈ ω)
50 2onn 7665 . . . . . . . . . . . 12 2𝑜 ∈ ω
51 nnmsuc 7632 . . . . . . . . . . . 12 ((2𝑜 ∈ ω ∧ 𝑦 ∈ ω) → (2𝑜 ·𝑜 suc 𝑦) = ((2𝑜 ·𝑜 𝑦) +𝑜 2𝑜))
5250, 51mpan 705 . . . . . . . . . . 11 (𝑦 ∈ ω → (2𝑜 ·𝑜 suc 𝑦) = ((2𝑜 ·𝑜 𝑦) +𝑜 2𝑜))
53 df-2o 7506 . . . . . . . . . . . . 13 2𝑜 = suc 1𝑜
5453oveq2i 6615 . . . . . . . . . . . 12 ((2𝑜 ·𝑜 𝑦) +𝑜 2𝑜) = ((2𝑜 ·𝑜 𝑦) +𝑜 suc 1𝑜)
55 nnmcl 7637 . . . . . . . . . . . . . 14 ((2𝑜 ∈ ω ∧ 𝑦 ∈ ω) → (2𝑜 ·𝑜 𝑦) ∈ ω)
5650, 55mpan 705 . . . . . . . . . . . . 13 (𝑦 ∈ ω → (2𝑜 ·𝑜 𝑦) ∈ ω)
57 1onn 7664 . . . . . . . . . . . . 13 1𝑜 ∈ ω
58 nnasuc 7631 . . . . . . . . . . . . 13 (((2𝑜 ·𝑜 𝑦) ∈ ω ∧ 1𝑜 ∈ ω) → ((2𝑜 ·𝑜 𝑦) +𝑜 suc 1𝑜) = suc ((2𝑜 ·𝑜 𝑦) +𝑜 1𝑜))
5956, 57, 58sylancl 693 . . . . . . . . . . . 12 (𝑦 ∈ ω → ((2𝑜 ·𝑜 𝑦) +𝑜 suc 1𝑜) = suc ((2𝑜 ·𝑜 𝑦) +𝑜 1𝑜))
6054, 59syl5req 2668 . . . . . . . . . . 11 (𝑦 ∈ ω → suc ((2𝑜 ·𝑜 𝑦) +𝑜 1𝑜) = ((2𝑜 ·𝑜 𝑦) +𝑜 2𝑜))
61 nnon 7018 . . . . . . . . . . . 12 ((2𝑜 ·𝑜 𝑦) ∈ ω → (2𝑜 ·𝑜 𝑦) ∈ On)
62 oa1suc 7556 . . . . . . . . . . . 12 ((2𝑜 ·𝑜 𝑦) ∈ On → ((2𝑜 ·𝑜 𝑦) +𝑜 1𝑜) = suc (2𝑜 ·𝑜 𝑦))
63 suceq 5749 . . . . . . . . . . . 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 2662 . . . . . . . . . 10 (𝑦 ∈ ω → suc suc (2𝑜 ·𝑜 𝑦) = (2𝑜 ·𝑜 suc 𝑦))
66 oveq2 6612 . . . . . . . . . . . 12 (𝑥 = suc 𝑦 → (2𝑜 ·𝑜 𝑥) = (2𝑜 ·𝑜 suc 𝑦))
6766eqeq2d 2631 . . . . . . . . . . 11 (𝑥 = suc 𝑦 → (suc suc (2𝑜 ·𝑜 𝑦) = (2𝑜 ·𝑜 𝑥) ↔ suc suc (2𝑜 ·𝑜 𝑦) = (2𝑜 ·𝑜 suc 𝑦)))
6867rspcev 3295 . . . . . . . . . 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 692 . . . . . . . . 9 (𝑦 ∈ ω → ∃𝑥 ∈ ω suc suc (2𝑜 ·𝑜 𝑦) = (2𝑜 ·𝑜 𝑥))
70 suceq 5749 . . . . . . . . . . . 12 (𝑧 = (2𝑜 ·𝑜 𝑦) → suc 𝑧 = suc (2𝑜 ·𝑜 𝑦))
71 suceq 5749 . . . . . . . . . . . 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 2623 . . . . . . . . . 10 (𝑧 = (2𝑜 ·𝑜 𝑦) → (suc suc 𝑧 = (2𝑜 ·𝑜 𝑥) ↔ suc suc (2𝑜 ·𝑜 𝑦) = (2𝑜 ·𝑜 𝑥)))
7473rexbidv 3045 . . . . . . . . 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 3020 . . . . . . 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 148 . . . 4 (𝑧 ∈ ω → (¬ ∃𝑥 ∈ ω suc suc 𝑧 = (2𝑜 ·𝑜 𝑥) → ¬ ∃𝑥 ∈ ω 𝑧 = (2𝑜 ·𝑜 𝑥)))
80 con1 143 . . . 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 7039 . 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 384   = wceq 1480  wcel 1987  wrex 2908  c0 3891  Oncon0 5682  suc csuc 5684  (class class class)co 6604  ωcom 7012  1𝑜c1o 7498  2𝑜c2o 7499   +𝑜 coa 7502   ·𝑜 comu 7503
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-2o 7506  df-oadd 7509  df-omul 7510
This theorem is referenced by:  fin1a2lem5  9170
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