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Theorem nneob 7619
 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 6557 . . . . 5 (𝑥 = 𝑦 → (2𝑜 ·𝑜 𝑥) = (2𝑜 ·𝑜 𝑦))
21eqeq2d 2620 . . . 4 (𝑥 = 𝑦 → (𝐴 = (2𝑜 ·𝑜 𝑥) ↔ 𝐴 = (2𝑜 ·𝑜 𝑦)))
32cbvrexv 3148 . . 3 (∃𝑥 ∈ ω 𝐴 = (2𝑜 ·𝑜 𝑥) ↔ ∃𝑦 ∈ ω 𝐴 = (2𝑜 ·𝑜 𝑦))
4 nnneo 7618 . . . . . . 7 ((𝑦 ∈ ω ∧ 𝑥 ∈ ω ∧ 𝐴 = (2𝑜 ·𝑜 𝑦)) → ¬ suc 𝐴 = (2𝑜 ·𝑜 𝑥))
543com23 1263 . . . . . 6 ((𝑦 ∈ ω ∧ 𝐴 = (2𝑜 ·𝑜 𝑦) ∧ 𝑥 ∈ ω) → ¬ suc 𝐴 = (2𝑜 ·𝑜 𝑥))
653expa 1257 . . . . 5 (((𝑦 ∈ ω ∧ 𝐴 = (2𝑜 ·𝑜 𝑦)) ∧ 𝑥 ∈ ω) → ¬ suc 𝐴 = (2𝑜 ·𝑜 𝑥))
76nrexdv 2984 . . . 4 ((𝑦 ∈ ω ∧ 𝐴 = (2𝑜 ·𝑜 𝑦)) → ¬ ∃𝑥 ∈ ω suc 𝐴 = (2𝑜 ·𝑜 𝑥))
87rexlimiva 3010 . . 3 (∃𝑦 ∈ ω 𝐴 = (2𝑜 ·𝑜 𝑦) → ¬ ∃𝑥 ∈ ω suc 𝐴 = (2𝑜 ·𝑜 𝑥))
93, 8sylbi 206 . 2 (∃𝑥 ∈ ω 𝐴 = (2𝑜 ·𝑜 𝑥) → ¬ ∃𝑥 ∈ ω suc 𝐴 = (2𝑜 ·𝑜 𝑥))
10 suceq 5707 . . . . . . 7 (𝑦 = ∅ → suc 𝑦 = suc ∅)
1110eqeq1d 2612 . . . . . 6 (𝑦 = ∅ → (suc 𝑦 = (2𝑜 ·𝑜 𝑥) ↔ suc ∅ = (2𝑜 ·𝑜 𝑥)))
1211rexbidv 3034 . . . . 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 2614 . . . . 5 (𝑦 = ∅ → (𝑦 = (2𝑜 ·𝑜 𝑥) ↔ ∅ = (2𝑜 ·𝑜 𝑥)))
1514rexbidv 3034 . . . 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 5707 . . . . . . 7 (𝑦 = 𝑧 → suc 𝑦 = suc 𝑧)
1817eqeq1d 2612 . . . . . 6 (𝑦 = 𝑧 → (suc 𝑦 = (2𝑜 ·𝑜 𝑥) ↔ suc 𝑧 = (2𝑜 ·𝑜 𝑥)))
1918rexbidv 3034 . . . . 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 2614 . . . . 5 (𝑦 = 𝑧 → (𝑦 = (2𝑜 ·𝑜 𝑥) ↔ 𝑧 = (2𝑜 ·𝑜 𝑥)))
2221rexbidv 3034 . . . 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 5707 . . . . . . 7 (𝑦 = suc 𝑧 → suc 𝑦 = suc suc 𝑧)
2524eqeq1d 2612 . . . . . 6 (𝑦 = suc 𝑧 → (suc 𝑦 = (2𝑜 ·𝑜 𝑥) ↔ suc suc 𝑧 = (2𝑜 ·𝑜 𝑥)))
2625rexbidv 3034 . . . . 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 2614 . . . . 5 (𝑦 = suc 𝑧 → (𝑦 = (2𝑜 ·𝑜 𝑥) ↔ suc 𝑧 = (2𝑜 ·𝑜 𝑥)))
2928rexbidv 3034 . . . 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 5707 . . . . . . 7 (𝑦 = 𝐴 → suc 𝑦 = suc 𝐴)
3231eqeq1d 2612 . . . . . 6 (𝑦 = 𝐴 → (suc 𝑦 = (2𝑜 ·𝑜 𝑥) ↔ suc 𝐴 = (2𝑜 ·𝑜 𝑥)))
3332rexbidv 3034 . . . . 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 2614 . . . . 5 (𝑦 = 𝐴 → (𝑦 = (2𝑜 ·𝑜 𝑥) ↔ 𝐴 = (2𝑜 ·𝑜 𝑥)))
3635rexbidv 3034 . . . 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 6977 . . . . 5 ∅ ∈ ω
39 eqid 2610 . . . . 5 ∅ = ∅
40 oveq2 6557 . . . . . . . 8 (𝑥 = ∅ → (2𝑜 ·𝑜 𝑥) = (2𝑜 ·𝑜 ∅))
41 om0x 7486 . . . . . . . 8 (2𝑜 ·𝑜 ∅) = ∅
4240, 41syl6eq 2660 . . . . . . 7 (𝑥 = ∅ → (2𝑜 ·𝑜 𝑥) = ∅)
4342eqeq2d 2620 . . . . . 6 (𝑥 = ∅ → (∅ = (2𝑜 ·𝑜 𝑥) ↔ ∅ = ∅))
4443rspcev 3282 . . . . 5 ((∅ ∈ ω ∧ ∅ = ∅) → ∃𝑥 ∈ ω ∅ = (2𝑜 ·𝑜 𝑥))
4538, 39, 44mp2an 704 . . . 4 𝑥 ∈ ω ∅ = (2𝑜 ·𝑜 𝑥)
4645a1i 11 . . 3 (¬ ∃𝑥 ∈ ω suc ∅ = (2𝑜 ·𝑜 𝑥) → ∃𝑥 ∈ ω ∅ = (2𝑜 ·𝑜 𝑥))
471eqeq2d 2620 . . . . . . 7 (𝑥 = 𝑦 → (𝑧 = (2𝑜 ·𝑜 𝑥) ↔ 𝑧 = (2𝑜 ·𝑜 𝑦)))
4847cbvrexv 3148 . . . . . 6 (∃𝑥 ∈ ω 𝑧 = (2𝑜 ·𝑜 𝑥) ↔ ∃𝑦 ∈ ω 𝑧 = (2𝑜 ·𝑜 𝑦))
49 peano2 6978 . . . . . . . . . 10 (𝑦 ∈ ω → suc 𝑦 ∈ ω)
50 2onn 7607 . . . . . . . . . . . 12 2𝑜 ∈ ω
51 nnmsuc 7574 . . . . . . . . . . . 12 ((2𝑜 ∈ ω ∧ 𝑦 ∈ ω) → (2𝑜 ·𝑜 suc 𝑦) = ((2𝑜 ·𝑜 𝑦) +𝑜 2𝑜))
5250, 51mpan 702 . . . . . . . . . . 11 (𝑦 ∈ ω → (2𝑜 ·𝑜 suc 𝑦) = ((2𝑜 ·𝑜 𝑦) +𝑜 2𝑜))
53 df-2o 7448 . . . . . . . . . . . . 13 2𝑜 = suc 1𝑜
5453oveq2i 6560 . . . . . . . . . . . 12 ((2𝑜 ·𝑜 𝑦) +𝑜 2𝑜) = ((2𝑜 ·𝑜 𝑦) +𝑜 suc 1𝑜)
55 nnmcl 7579 . . . . . . . . . . . . . 14 ((2𝑜 ∈ ω ∧ 𝑦 ∈ ω) → (2𝑜 ·𝑜 𝑦) ∈ ω)
5650, 55mpan 702 . . . . . . . . . . . . 13 (𝑦 ∈ ω → (2𝑜 ·𝑜 𝑦) ∈ ω)
57 1onn 7606 . . . . . . . . . . . . 13 1𝑜 ∈ ω
58 nnasuc 7573 . . . . . . . . . . . . 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 2657 . . . . . . . . . . 11 (𝑦 ∈ ω → suc ((2𝑜 ·𝑜 𝑦) +𝑜 1𝑜) = ((2𝑜 ·𝑜 𝑦) +𝑜 2𝑜))
61 nnon 6963 . . . . . . . . . . . 12 ((2𝑜 ·𝑜 𝑦) ∈ ω → (2𝑜 ·𝑜 𝑦) ∈ On)
62 oa1suc 7498 . . . . . . . . . . . 12 ((2𝑜 ·𝑜 𝑦) ∈ On → ((2𝑜 ·𝑜 𝑦) +𝑜 1𝑜) = suc (2𝑜 ·𝑜 𝑦))
63 suceq 5707 . . . . . . . . . . . 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 2651 . . . . . . . . . 10 (𝑦 ∈ ω → suc suc (2𝑜 ·𝑜 𝑦) = (2𝑜 ·𝑜 suc 𝑦))
66 oveq2 6557 . . . . . . . . . . . 12 (𝑥 = suc 𝑦 → (2𝑜 ·𝑜 𝑥) = (2𝑜 ·𝑜 suc 𝑦))
6766eqeq2d 2620 . . . . . . . . . . 11 (𝑥 = suc 𝑦 → (suc suc (2𝑜 ·𝑜 𝑦) = (2𝑜 ·𝑜 𝑥) ↔ suc suc (2𝑜 ·𝑜 𝑦) = (2𝑜 ·𝑜 suc 𝑦)))
6867rspcev 3282 . . . . . . . . . 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 691 . . . . . . . . 9 (𝑦 ∈ ω → ∃𝑥 ∈ ω suc suc (2𝑜 ·𝑜 𝑦) = (2𝑜 ·𝑜 𝑥))
70 suceq 5707 . . . . . . . . . . . 12 (𝑧 = (2𝑜 ·𝑜 𝑦) → suc 𝑧 = suc (2𝑜 ·𝑜 𝑦))
71 suceq 5707 . . . . . . . . . . . 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 2612 . . . . . . . . . 10 (𝑧 = (2𝑜 ·𝑜 𝑦) → (suc suc 𝑧 = (2𝑜 ·𝑜 𝑥) ↔ suc suc (2𝑜 ·𝑜 𝑦) = (2𝑜 ·𝑜 𝑥)))
7473rexbidv 3034 . . . . . . . . 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 236 . . . . . . . 8 (𝑦 ∈ ω → (𝑧 = (2𝑜 ·𝑜 𝑦) → ∃𝑥 ∈ ω suc suc 𝑧 = (2𝑜 ·𝑜 𝑥)))
7675rexlimiv 3009 . . . . . . 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 231 . . . . 5 (𝑧 ∈ ω → (∃𝑥 ∈ ω 𝑧 = (2𝑜 ·𝑜 𝑥) → ∃𝑥 ∈ ω suc suc 𝑧 = (2𝑜 ·𝑜 𝑥)))
7978con3d 147 . . . 4 (𝑧 ∈ ω → (¬ ∃𝑥 ∈ ω suc suc 𝑧 = (2𝑜 ·𝑜 𝑥) → ¬ ∃𝑥 ∈ ω 𝑧 = (2𝑜 ·𝑜 𝑥)))
80 con1 142 . . . 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 75 . . 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 6984 . 2 (𝐴 ∈ ω → (¬ ∃𝑥 ∈ ω suc 𝐴 = (2𝑜 ·𝑜 𝑥) → ∃𝑥 ∈ ω 𝐴 = (2𝑜 ·𝑜 𝑥)))
839, 82impbid2 215 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 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∃wrex 2897  ∅c0 3874  Oncon0 5640  suc csuc 5642  (class class class)co 6549  ωcom 6957  1𝑜c1o 7440  2𝑜c2o 7441   +𝑜 coa 7444   ·𝑜 comu 7445 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-omul 7452 This theorem is referenced by:  fin1a2lem5  9109
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