Proof of Theorem nnneo
| Step | Hyp | Ref
| Expression |
| 1 | | nnon 7893 |
. . . 4
⊢ (𝐴 ∈ ω → 𝐴 ∈ On) |
| 2 | | onnbtwn 6478 |
. . . 4
⊢ (𝐴 ∈ On → ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ suc 𝐴)) |
| 3 | 1, 2 | syl 17 |
. . 3
⊢ (𝐴 ∈ ω → ¬
(𝐴 ∈ 𝐵 ∧ 𝐵 ∈ suc 𝐴)) |
| 4 | 3 | 3ad2ant1 1134 |
. 2
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 = (2o
·o 𝐴))
→ ¬ (𝐴 ∈
𝐵 ∧ 𝐵 ∈ suc 𝐴)) |
| 5 | | suceq 6450 |
. . . . 5
⊢ (𝐶 = (2o
·o 𝐴)
→ suc 𝐶 = suc
(2o ·o 𝐴)) |
| 6 | 5 | eqeq1d 2739 |
. . . 4
⊢ (𝐶 = (2o
·o 𝐴)
→ (suc 𝐶 =
(2o ·o 𝐵) ↔ suc (2o
·o 𝐴) =
(2o ·o 𝐵))) |
| 7 | 6 | 3ad2ant3 1136 |
. . 3
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 = (2o
·o 𝐴))
→ (suc 𝐶 =
(2o ·o 𝐵) ↔ suc (2o
·o 𝐴) =
(2o ·o 𝐵))) |
| 8 | | ovex 7464 |
. . . . . . . 8
⊢
(2o ·o 𝐴) ∈ V |
| 9 | 8 | sucid 6466 |
. . . . . . 7
⊢
(2o ·o 𝐴) ∈ suc (2o
·o 𝐴) |
| 10 | | eleq2 2830 |
. . . . . . 7
⊢ (suc
(2o ·o 𝐴) = (2o ·o
𝐵) → ((2o
·o 𝐴)
∈ suc (2o ·o 𝐴) ↔ (2o ·o
𝐴) ∈ (2o
·o 𝐵))) |
| 11 | 9, 10 | mpbii 233 |
. . . . . 6
⊢ (suc
(2o ·o 𝐴) = (2o ·o
𝐵) → (2o
·o 𝐴)
∈ (2o ·o 𝐵)) |
| 12 | | 2onn 8680 |
. . . . . . . 8
⊢
2o ∈ ω |
| 13 | | nnmord 8670 |
. . . . . . . 8
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧
2o ∈ ω) → ((𝐴 ∈ 𝐵 ∧ ∅ ∈ 2o) ↔
(2o ·o 𝐴) ∈ (2o ·o
𝐵))) |
| 14 | 12, 13 | mp3an3 1452 |
. . . . . . 7
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐴 ∈ 𝐵 ∧ ∅ ∈ 2o) ↔
(2o ·o 𝐴) ∈ (2o ·o
𝐵))) |
| 15 | | simpl 482 |
. . . . . . 7
⊢ ((𝐴 ∈ 𝐵 ∧ ∅ ∈ 2o) →
𝐴 ∈ 𝐵) |
| 16 | 14, 15 | biimtrrdi 254 |
. . . . . 6
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) →
((2o ·o 𝐴) ∈ (2o ·o
𝐵) → 𝐴 ∈ 𝐵)) |
| 17 | 11, 16 | syl5 34 |
. . . . 5
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (suc
(2o ·o 𝐴) = (2o ·o
𝐵) → 𝐴 ∈ 𝐵)) |
| 18 | | simpr 484 |
. . . . . . . . 9
⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ suc
(2o ·o 𝐴) = (2o ·o
𝐵)) → suc
(2o ·o 𝐴) = (2o ·o
𝐵)) |
| 19 | | nnmcl 8650 |
. . . . . . . . . . . . 13
⊢
((2o ∈ ω ∧ 𝐴 ∈ ω) → (2o
·o 𝐴)
∈ ω) |
| 20 | 12, 19 | mpan 690 |
. . . . . . . . . . . 12
⊢ (𝐴 ∈ ω →
(2o ·o 𝐴) ∈ ω) |
| 21 | | nnon 7893 |
. . . . . . . . . . . 12
⊢
((2o ·o 𝐴) ∈ ω → (2o
·o 𝐴)
∈ On) |
| 22 | | oa1suc 8569 |
. . . . . . . . . . . 12
⊢
((2o ·o 𝐴) ∈ On → ((2o
·o 𝐴)
+o 1o) = suc (2o ·o 𝐴)) |
| 23 | 20, 21, 22 | 3syl 18 |
. . . . . . . . . . 11
⊢ (𝐴 ∈ ω →
((2o ·o 𝐴) +o 1o) = suc
(2o ·o 𝐴)) |
| 24 | | 1oex 8516 |
. . . . . . . . . . . . . . 15
⊢
1o ∈ V |
| 25 | 24 | sucid 6466 |
. . . . . . . . . . . . . 14
⊢
1o ∈ suc 1o |
| 26 | | df-2o 8507 |
. . . . . . . . . . . . . 14
⊢
2o = suc 1o |
| 27 | 25, 26 | eleqtrri 2840 |
. . . . . . . . . . . . 13
⊢
1o ∈ 2o |
| 28 | | 1onn 8678 |
. . . . . . . . . . . . . 14
⊢
1o ∈ ω |
| 29 | | nnaord 8657 |
. . . . . . . . . . . . . 14
⊢
((1o ∈ ω ∧ 2o ∈ ω ∧
(2o ·o 𝐴) ∈ ω) → (1o
∈ 2o ↔ ((2o ·o 𝐴) +o 1o)
∈ ((2o ·o 𝐴) +o
2o))) |
| 30 | 28, 12, 20, 29 | mp3an12i 1467 |
. . . . . . . . . . . . 13
⊢ (𝐴 ∈ ω →
(1o ∈ 2o ↔ ((2o ·o
𝐴) +o
1o) ∈ ((2o ·o 𝐴) +o
2o))) |
| 31 | 27, 30 | mpbii 233 |
. . . . . . . . . . . 12
⊢ (𝐴 ∈ ω →
((2o ·o 𝐴) +o 1o) ∈
((2o ·o 𝐴) +o
2o)) |
| 32 | | nnmsuc 8645 |
. . . . . . . . . . . . 13
⊢
((2o ∈ ω ∧ 𝐴 ∈ ω) → (2o
·o suc 𝐴)
= ((2o ·o 𝐴) +o
2o)) |
| 33 | 12, 32 | mpan 690 |
. . . . . . . . . . . 12
⊢ (𝐴 ∈ ω →
(2o ·o suc 𝐴) = ((2o ·o
𝐴) +o
2o)) |
| 34 | 31, 33 | eleqtrrd 2844 |
. . . . . . . . . . 11
⊢ (𝐴 ∈ ω →
((2o ·o 𝐴) +o 1o) ∈
(2o ·o suc 𝐴)) |
| 35 | 23, 34 | eqeltrrd 2842 |
. . . . . . . . . 10
⊢ (𝐴 ∈ ω → suc
(2o ·o 𝐴) ∈ (2o ·o
suc 𝐴)) |
| 36 | 35 | ad2antrr 726 |
. . . . . . . . 9
⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ suc
(2o ·o 𝐴) = (2o ·o
𝐵)) → suc
(2o ·o 𝐴) ∈ (2o ·o
suc 𝐴)) |
| 37 | 18, 36 | eqeltrrd 2842 |
. . . . . . . 8
⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ suc
(2o ·o 𝐴) = (2o ·o
𝐵)) → (2o
·o 𝐵)
∈ (2o ·o suc 𝐴)) |
| 38 | | peano2 7912 |
. . . . . . . . . . 11
⊢ (𝐴 ∈ ω → suc 𝐴 ∈
ω) |
| 39 | | nnmord 8670 |
. . . . . . . . . . . 12
⊢ ((𝐵 ∈ ω ∧ suc 𝐴 ∈ ω ∧
2o ∈ ω) → ((𝐵 ∈ suc 𝐴 ∧ ∅ ∈ 2o) ↔
(2o ·o 𝐵) ∈ (2o ·o
suc 𝐴))) |
| 40 | 12, 39 | mp3an3 1452 |
. . . . . . . . . . 11
⊢ ((𝐵 ∈ ω ∧ suc 𝐴 ∈ ω) → ((𝐵 ∈ suc 𝐴 ∧ ∅ ∈ 2o) ↔
(2o ·o 𝐵) ∈ (2o ·o
suc 𝐴))) |
| 41 | 38, 40 | sylan2 593 |
. . . . . . . . . 10
⊢ ((𝐵 ∈ ω ∧ 𝐴 ∈ ω) → ((𝐵 ∈ suc 𝐴 ∧ ∅ ∈ 2o) ↔
(2o ·o 𝐵) ∈ (2o ·o
suc 𝐴))) |
| 42 | 41 | ancoms 458 |
. . . . . . . . 9
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐵 ∈ suc 𝐴 ∧ ∅ ∈ 2o) ↔
(2o ·o 𝐵) ∈ (2o ·o
suc 𝐴))) |
| 43 | 42 | adantr 480 |
. . . . . . . 8
⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ suc
(2o ·o 𝐴) = (2o ·o
𝐵)) → ((𝐵 ∈ suc 𝐴 ∧ ∅ ∈ 2o) ↔
(2o ·o 𝐵) ∈ (2o ·o
suc 𝐴))) |
| 44 | 37, 43 | mpbird 257 |
. . . . . . 7
⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ suc
(2o ·o 𝐴) = (2o ·o
𝐵)) → (𝐵 ∈ suc 𝐴 ∧ ∅ ∈
2o)) |
| 45 | 44 | simpld 494 |
. . . . . 6
⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ suc
(2o ·o 𝐴) = (2o ·o
𝐵)) → 𝐵 ∈ suc 𝐴) |
| 46 | 45 | ex 412 |
. . . . 5
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (suc
(2o ·o 𝐴) = (2o ·o
𝐵) → 𝐵 ∈ suc 𝐴)) |
| 47 | 17, 46 | jcad 512 |
. . . 4
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (suc
(2o ·o 𝐴) = (2o ·o
𝐵) → (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ suc 𝐴))) |
| 48 | 47 | 3adant3 1133 |
. . 3
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 = (2o
·o 𝐴))
→ (suc (2o ·o 𝐴) = (2o ·o
𝐵) → (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ suc 𝐴))) |
| 49 | 7, 48 | sylbid 240 |
. 2
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 = (2o
·o 𝐴))
→ (suc 𝐶 =
(2o ·o 𝐵) → (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ suc 𝐴))) |
| 50 | 4, 49 | mtod 198 |
1
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 = (2o
·o 𝐴))
→ ¬ suc 𝐶 =
(2o ·o 𝐵)) |