Proof of Theorem nosepssdm
Step | Hyp | Ref
| Expression |
1 | | nosepne 32344 |
. . . 4
⊢ ((𝐴 ∈
No ∧ 𝐵 ∈
No ∧ 𝐴 ≠ 𝐵) → (𝐴‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) ≠ (𝐵‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)})) |
2 | 1 | neneqd 2976 |
. . 3
⊢ ((𝐴 ∈
No ∧ 𝐵 ∈
No ∧ 𝐴 ≠ 𝐵) → ¬ (𝐴‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) = (𝐵‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)})) |
3 | | nodmord 32319 |
. . . . . . . . 9
⊢ (𝐴 ∈
No → Ord dom 𝐴) |
4 | 3 | 3ad2ant1 1164 |
. . . . . . . 8
⊢ ((𝐴 ∈
No ∧ 𝐵 ∈
No ∧ 𝐴 ≠ 𝐵) → Ord dom 𝐴) |
5 | | ordn2lp 5961 |
. . . . . . . 8
⊢ (Ord dom
𝐴 → ¬ (dom 𝐴 ∈ ∩ {𝑥
∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)} ∧ ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)} ∈ dom 𝐴)) |
6 | 4, 5 | syl 17 |
. . . . . . 7
⊢ ((𝐴 ∈
No ∧ 𝐵 ∈
No ∧ 𝐴 ≠ 𝐵) → ¬ (dom 𝐴 ∈ ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)} ∧ ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)} ∈ dom 𝐴)) |
7 | | imnan 389 |
. . . . . . 7
⊢ ((dom
𝐴 ∈ ∩ {𝑥
∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)} → ¬ ∩
{𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)} ∈ dom 𝐴) ↔ ¬ (dom 𝐴 ∈ ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)} ∧ ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)} ∈ dom 𝐴)) |
8 | 6, 7 | sylibr 226 |
. . . . . 6
⊢ ((𝐴 ∈
No ∧ 𝐵 ∈
No ∧ 𝐴 ≠ 𝐵) → (dom 𝐴 ∈ ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)} → ¬ ∩
{𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)} ∈ dom 𝐴)) |
9 | 8 | imp 396 |
. . . . 5
⊢ (((𝐴 ∈
No ∧ 𝐵 ∈
No ∧ 𝐴 ≠ 𝐵) ∧ dom 𝐴 ∈ ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) → ¬ ∩
{𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)} ∈ dom 𝐴) |
10 | | ndmfv 6441 |
. . . . 5
⊢ (¬
∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)} ∈ dom 𝐴 → (𝐴‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) = ∅) |
11 | 9, 10 | syl 17 |
. . . 4
⊢ (((𝐴 ∈
No ∧ 𝐵 ∈
No ∧ 𝐴 ≠ 𝐵) ∧ dom 𝐴 ∈ ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) → (𝐴‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) = ∅) |
12 | | nosepeq 32348 |
. . . . . 6
⊢ (((𝐴 ∈
No ∧ 𝐵 ∈
No ∧ 𝐴 ≠ 𝐵) ∧ dom 𝐴 ∈ ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) → (𝐴‘dom 𝐴) = (𝐵‘dom 𝐴)) |
13 | | simpl1 1243 |
. . . . . . . . . . 11
⊢ (((𝐴 ∈
No ∧ 𝐵 ∈
No ∧ 𝐴 ≠ 𝐵) ∧ dom 𝐴 ∈ ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) → 𝐴 ∈ No
) |
14 | 13, 3 | syl 17 |
. . . . . . . . . 10
⊢ (((𝐴 ∈
No ∧ 𝐵 ∈
No ∧ 𝐴 ≠ 𝐵) ∧ dom 𝐴 ∈ ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) → Ord dom 𝐴) |
15 | | ordirr 5959 |
. . . . . . . . . 10
⊢ (Ord dom
𝐴 → ¬ dom 𝐴 ∈ dom 𝐴) |
16 | | ndmfv 6441 |
. . . . . . . . . 10
⊢ (¬
dom 𝐴 ∈ dom 𝐴 → (𝐴‘dom 𝐴) = ∅) |
17 | 14, 15, 16 | 3syl 18 |
. . . . . . . . 9
⊢ (((𝐴 ∈
No ∧ 𝐵 ∈
No ∧ 𝐴 ≠ 𝐵) ∧ dom 𝐴 ∈ ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) → (𝐴‘dom 𝐴) = ∅) |
18 | 17 | eqeq1d 2801 |
. . . . . . . 8
⊢ (((𝐴 ∈
No ∧ 𝐵 ∈
No ∧ 𝐴 ≠ 𝐵) ∧ dom 𝐴 ∈ ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) → ((𝐴‘dom 𝐴) = (𝐵‘dom 𝐴) ↔ ∅ = (𝐵‘dom 𝐴))) |
19 | | eqcom 2806 |
. . . . . . . 8
⊢ (∅
= (𝐵‘dom 𝐴) ↔ (𝐵‘dom 𝐴) = ∅) |
20 | 18, 19 | syl6bb 279 |
. . . . . . 7
⊢ (((𝐴 ∈
No ∧ 𝐵 ∈
No ∧ 𝐴 ≠ 𝐵) ∧ dom 𝐴 ∈ ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) → ((𝐴‘dom 𝐴) = (𝐵‘dom 𝐴) ↔ (𝐵‘dom 𝐴) = ∅)) |
21 | | simpl2 1245 |
. . . . . . . . . . 11
⊢ (((𝐴 ∈
No ∧ 𝐵 ∈
No ∧ 𝐴 ≠ 𝐵) ∧ dom 𝐴 ∈ ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) → 𝐵 ∈ No
) |
22 | | nofun 32315 |
. . . . . . . . . . 11
⊢ (𝐵 ∈
No → Fun 𝐵) |
23 | 21, 22 | syl 17 |
. . . . . . . . . 10
⊢ (((𝐴 ∈
No ∧ 𝐵 ∈
No ∧ 𝐴 ≠ 𝐵) ∧ dom 𝐴 ∈ ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) → Fun 𝐵) |
24 | | nosgnn0 32324 |
. . . . . . . . . . 11
⊢ ¬
∅ ∈ {1𝑜, 2𝑜} |
25 | | norn 32317 |
. . . . . . . . . . . . 13
⊢ (𝐵 ∈
No → ran 𝐵
⊆ {1𝑜, 2𝑜}) |
26 | 21, 25 | syl 17 |
. . . . . . . . . . . 12
⊢ (((𝐴 ∈
No ∧ 𝐵 ∈
No ∧ 𝐴 ≠ 𝐵) ∧ dom 𝐴 ∈ ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) → ran 𝐵 ⊆ {1𝑜,
2𝑜}) |
27 | 26 | sseld 3797 |
. . . . . . . . . . 11
⊢ (((𝐴 ∈
No ∧ 𝐵 ∈
No ∧ 𝐴 ≠ 𝐵) ∧ dom 𝐴 ∈ ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) → (∅ ∈ ran 𝐵 → ∅ ∈
{1𝑜, 2𝑜})) |
28 | 24, 27 | mtoi 191 |
. . . . . . . . . 10
⊢ (((𝐴 ∈
No ∧ 𝐵 ∈
No ∧ 𝐴 ≠ 𝐵) ∧ dom 𝐴 ∈ ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) → ¬ ∅ ∈ ran 𝐵) |
29 | | funeldmb 32175 |
. . . . . . . . . 10
⊢ ((Fun
𝐵 ∧ ¬ ∅
∈ ran 𝐵) → (dom
𝐴 ∈ dom 𝐵 ↔ (𝐵‘dom 𝐴) ≠ ∅)) |
30 | 23, 28, 29 | syl2anc 580 |
. . . . . . . . 9
⊢ (((𝐴 ∈
No ∧ 𝐵 ∈
No ∧ 𝐴 ≠ 𝐵) ∧ dom 𝐴 ∈ ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) → (dom 𝐴 ∈ dom 𝐵 ↔ (𝐵‘dom 𝐴) ≠ ∅)) |
31 | 30 | necon2bbid 3014 |
. . . . . . . 8
⊢ (((𝐴 ∈
No ∧ 𝐵 ∈
No ∧ 𝐴 ≠ 𝐵) ∧ dom 𝐴 ∈ ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) → ((𝐵‘dom 𝐴) = ∅ ↔ ¬ dom 𝐴 ∈ dom 𝐵)) |
32 | | nodmord 32319 |
. . . . . . . . . . . 12
⊢ (𝐵 ∈
No → Ord dom 𝐵) |
33 | 32 | 3ad2ant2 1165 |
. . . . . . . . . . 11
⊢ ((𝐴 ∈
No ∧ 𝐵 ∈
No ∧ 𝐴 ≠ 𝐵) → Ord dom 𝐵) |
34 | | ordtr1 5984 |
. . . . . . . . . . 11
⊢ (Ord dom
𝐵 → ((dom 𝐴 ∈ ∩ {𝑥
∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)} ∧ ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)} ∈ dom 𝐵) → dom 𝐴 ∈ dom 𝐵)) |
35 | 33, 34 | syl 17 |
. . . . . . . . . 10
⊢ ((𝐴 ∈
No ∧ 𝐵 ∈
No ∧ 𝐴 ≠ 𝐵) → ((dom 𝐴 ∈ ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)} ∧ ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)} ∈ dom 𝐵) → dom 𝐴 ∈ dom 𝐵)) |
36 | 35 | expdimp 445 |
. . . . . . . . 9
⊢ (((𝐴 ∈
No ∧ 𝐵 ∈
No ∧ 𝐴 ≠ 𝐵) ∧ dom 𝐴 ∈ ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) → (∩
{𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)} ∈ dom 𝐵 → dom 𝐴 ∈ dom 𝐵)) |
37 | 36 | con3d 150 |
. . . . . . . 8
⊢ (((𝐴 ∈
No ∧ 𝐵 ∈
No ∧ 𝐴 ≠ 𝐵) ∧ dom 𝐴 ∈ ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) → (¬ dom 𝐴 ∈ dom 𝐵 → ¬ ∩
{𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)} ∈ dom 𝐵)) |
38 | 31, 37 | sylbid 232 |
. . . . . . 7
⊢ (((𝐴 ∈
No ∧ 𝐵 ∈
No ∧ 𝐴 ≠ 𝐵) ∧ dom 𝐴 ∈ ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) → ((𝐵‘dom 𝐴) = ∅ → ¬ ∩ {𝑥
∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)} ∈ dom 𝐵)) |
39 | 20, 38 | sylbid 232 |
. . . . . 6
⊢ (((𝐴 ∈
No ∧ 𝐵 ∈
No ∧ 𝐴 ≠ 𝐵) ∧ dom 𝐴 ∈ ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) → ((𝐴‘dom 𝐴) = (𝐵‘dom 𝐴) → ¬ ∩
{𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)} ∈ dom 𝐵)) |
40 | 12, 39 | mpd 15 |
. . . . 5
⊢ (((𝐴 ∈
No ∧ 𝐵 ∈
No ∧ 𝐴 ≠ 𝐵) ∧ dom 𝐴 ∈ ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) → ¬ ∩
{𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)} ∈ dom 𝐵) |
41 | | ndmfv 6441 |
. . . . 5
⊢ (¬
∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)} ∈ dom 𝐵 → (𝐵‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) = ∅) |
42 | 40, 41 | syl 17 |
. . . 4
⊢ (((𝐴 ∈
No ∧ 𝐵 ∈
No ∧ 𝐴 ≠ 𝐵) ∧ dom 𝐴 ∈ ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) → (𝐵‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) = ∅) |
43 | 11, 42 | eqtr4d 2836 |
. . 3
⊢ (((𝐴 ∈
No ∧ 𝐵 ∈
No ∧ 𝐴 ≠ 𝐵) ∧ dom 𝐴 ∈ ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) → (𝐴‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) = (𝐵‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)})) |
44 | 2, 43 | mtand 851 |
. 2
⊢ ((𝐴 ∈
No ∧ 𝐵 ∈
No ∧ 𝐴 ≠ 𝐵) → ¬ dom 𝐴 ∈ ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) |
45 | | nosepon 32331 |
. . 3
⊢ ((𝐴 ∈
No ∧ 𝐵 ∈
No ∧ 𝐴 ≠ 𝐵) → ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)} ∈ On) |
46 | | nodmon 32316 |
. . . 4
⊢ (𝐴 ∈
No → dom 𝐴
∈ On) |
47 | 46 | 3ad2ant1 1164 |
. . 3
⊢ ((𝐴 ∈
No ∧ 𝐵 ∈
No ∧ 𝐴 ≠ 𝐵) → dom 𝐴 ∈ On) |
48 | | ontri1 5975 |
. . 3
⊢ ((∩ {𝑥
∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)} ∈ On ∧ dom 𝐴 ∈ On) → (∩ {𝑥
∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)} ⊆ dom 𝐴 ↔ ¬ dom 𝐴 ∈ ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)})) |
49 | 45, 47, 48 | syl2anc 580 |
. 2
⊢ ((𝐴 ∈
No ∧ 𝐵 ∈
No ∧ 𝐴 ≠ 𝐵) → (∩
{𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)} ⊆ dom 𝐴 ↔ ¬ dom 𝐴 ∈ ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)})) |
50 | 44, 49 | mpbird 249 |
1
⊢ ((𝐴 ∈
No ∧ 𝐵 ∈
No ∧ 𝐴 ≠ 𝐵) → ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)} ⊆ dom 𝐴) |