Proof of Theorem nosepssdm
Step | Hyp | Ref
| Expression |
1 | | nosepne 27707 |
. . . 4
⊢ ((𝐴 ∈
No ∧ 𝐵 ∈
No ∧ 𝐴 ≠ 𝐵) → (𝐴‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) ≠ (𝐵‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)})) |
2 | 1 | neneqd 2935 |
. . 3
⊢ ((𝐴 ∈
No ∧ 𝐵 ∈
No ∧ 𝐴 ≠ 𝐵) → ¬ (𝐴‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) = (𝐵‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)})) |
3 | | nodmord 27680 |
. . . . . . . . 9
⊢ (𝐴 ∈
No → Ord dom 𝐴) |
4 | 3 | 3ad2ant1 1130 |
. . . . . . . 8
⊢ ((𝐴 ∈
No ∧ 𝐵 ∈
No ∧ 𝐴 ≠ 𝐵) → Ord dom 𝐴) |
5 | | ordn2lp 6388 |
. . . . . . . 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 398 |
. . . . . . 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 233 |
. . . . . 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 405 |
. . . . 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 6928 |
. . . . 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 27712 |
. . . . . 6
⊢ (((𝐴 ∈
No ∧ 𝐵 ∈
No ∧ 𝐴 ≠ 𝐵) ∧ dom 𝐴 ∈ ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) → (𝐴‘dom 𝐴) = (𝐵‘dom 𝐴)) |
13 | | simpl1 1188 |
. . . . . . . . . 10
⊢ (((𝐴 ∈
No ∧ 𝐵 ∈
No ∧ 𝐴 ≠ 𝐵) ∧ dom 𝐴 ∈ ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) → 𝐴 ∈ No
) |
14 | | ordirr 6386 |
. . . . . . . . . 10
⊢ (Ord dom
𝐴 → ¬ dom 𝐴 ∈ dom 𝐴) |
15 | | ndmfv 6928 |
. . . . . . . . . 10
⊢ (¬
dom 𝐴 ∈ dom 𝐴 → (𝐴‘dom 𝐴) = ∅) |
16 | 13, 3, 14, 15 | 4syl 19 |
. . . . . . . . 9
⊢ (((𝐴 ∈
No ∧ 𝐵 ∈
No ∧ 𝐴 ≠ 𝐵) ∧ dom 𝐴 ∈ ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) → (𝐴‘dom 𝐴) = ∅) |
17 | 16 | eqeq1d 2728 |
. . . . . . . 8
⊢ (((𝐴 ∈
No ∧ 𝐵 ∈
No ∧ 𝐴 ≠ 𝐵) ∧ dom 𝐴 ∈ ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) → ((𝐴‘dom 𝐴) = (𝐵‘dom 𝐴) ↔ ∅ = (𝐵‘dom 𝐴))) |
18 | | eqcom 2733 |
. . . . . . . 8
⊢ (∅
= (𝐵‘dom 𝐴) ↔ (𝐵‘dom 𝐴) = ∅) |
19 | 17, 18 | bitrdi 286 |
. . . . . . 7
⊢ (((𝐴 ∈
No ∧ 𝐵 ∈
No ∧ 𝐴 ≠ 𝐵) ∧ dom 𝐴 ∈ ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) → ((𝐴‘dom 𝐴) = (𝐵‘dom 𝐴) ↔ (𝐵‘dom 𝐴) = ∅)) |
20 | | simpl2 1189 |
. . . . . . . . . . 11
⊢ (((𝐴 ∈
No ∧ 𝐵 ∈
No ∧ 𝐴 ≠ 𝐵) ∧ dom 𝐴 ∈ ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) → 𝐵 ∈ No
) |
21 | | nofun 27676 |
. . . . . . . . . . 11
⊢ (𝐵 ∈
No → Fun 𝐵) |
22 | 20, 21 | syl 17 |
. . . . . . . . . 10
⊢ (((𝐴 ∈
No ∧ 𝐵 ∈
No ∧ 𝐴 ≠ 𝐵) ∧ dom 𝐴 ∈ ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) → Fun 𝐵) |
23 | | nosgnn0 27685 |
. . . . . . . . . . 11
⊢ ¬
∅ ∈ {1o, 2o} |
24 | | norn 27678 |
. . . . . . . . . . . . 13
⊢ (𝐵 ∈
No → ran 𝐵
⊆ {1o, 2o}) |
25 | 20, 24 | syl 17 |
. . . . . . . . . . . 12
⊢ (((𝐴 ∈
No ∧ 𝐵 ∈
No ∧ 𝐴 ≠ 𝐵) ∧ dom 𝐴 ∈ ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) → ran 𝐵 ⊆ {1o,
2o}) |
26 | 25 | sseld 3977 |
. . . . . . . . . . 11
⊢ (((𝐴 ∈
No ∧ 𝐵 ∈
No ∧ 𝐴 ≠ 𝐵) ∧ dom 𝐴 ∈ ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) → (∅ ∈ ran 𝐵 → ∅ ∈
{1o, 2o})) |
27 | 23, 26 | mtoi 198 |
. . . . . . . . . 10
⊢ (((𝐴 ∈
No ∧ 𝐵 ∈
No ∧ 𝐴 ≠ 𝐵) ∧ dom 𝐴 ∈ ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) → ¬ ∅ ∈ ran 𝐵) |
28 | | funeldmb 7363 |
. . . . . . . . . 10
⊢ ((Fun
𝐵 ∧ ¬ ∅
∈ ran 𝐵) → (dom
𝐴 ∈ dom 𝐵 ↔ (𝐵‘dom 𝐴) ≠ ∅)) |
29 | 22, 27, 28 | syl2anc 582 |
. . . . . . . . 9
⊢ (((𝐴 ∈
No ∧ 𝐵 ∈
No ∧ 𝐴 ≠ 𝐵) ∧ dom 𝐴 ∈ ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) → (dom 𝐴 ∈ dom 𝐵 ↔ (𝐵‘dom 𝐴) ≠ ∅)) |
30 | 29 | necon2bbid 2974 |
. . . . . . . 8
⊢ (((𝐴 ∈
No ∧ 𝐵 ∈
No ∧ 𝐴 ≠ 𝐵) ∧ dom 𝐴 ∈ ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) → ((𝐵‘dom 𝐴) = ∅ ↔ ¬ dom 𝐴 ∈ dom 𝐵)) |
31 | | nodmord 27680 |
. . . . . . . . . . . 12
⊢ (𝐵 ∈
No → Ord dom 𝐵) |
32 | 31 | 3ad2ant2 1131 |
. . . . . . . . . . 11
⊢ ((𝐴 ∈
No ∧ 𝐵 ∈
No ∧ 𝐴 ≠ 𝐵) → Ord dom 𝐵) |
33 | | ordtr1 6411 |
. . . . . . . . . . 11
⊢ (Ord dom
𝐵 → ((dom 𝐴 ∈ ∩ {𝑥
∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)} ∧ ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)} ∈ dom 𝐵) → dom 𝐴 ∈ dom 𝐵)) |
34 | 32, 33 | 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)) |
35 | 34 | expdimp 451 |
. . . . . . . . 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)) |
36 | 35 | con3d 152 |
. . . . . . . 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)) |
37 | 30, 36 | sylbid 239 |
. . . . . . 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)) |
38 | 19, 37 | sylbid 239 |
. . . . . 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)) |
39 | 12, 38 | 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) |
40 | | ndmfv 6928 |
. . . . 5
⊢ (¬
∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)} ∈ dom 𝐵 → (𝐵‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) = ∅) |
41 | 39, 40 | syl 17 |
. . . 4
⊢ (((𝐴 ∈
No ∧ 𝐵 ∈
No ∧ 𝐴 ≠ 𝐵) ∧ dom 𝐴 ∈ ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) → (𝐵‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) = ∅) |
42 | 11, 41 | eqtr4d 2769 |
. . 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)})) |
43 | 2, 42 | mtand 814 |
. 2
⊢ ((𝐴 ∈
No ∧ 𝐵 ∈
No ∧ 𝐴 ≠ 𝐵) → ¬ dom 𝐴 ∈ ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) |
44 | | nosepon 27692 |
. . 3
⊢ ((𝐴 ∈
No ∧ 𝐵 ∈
No ∧ 𝐴 ≠ 𝐵) → ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)} ∈ On) |
45 | | nodmon 27677 |
. . . 4
⊢ (𝐴 ∈
No → dom 𝐴
∈ On) |
46 | 45 | 3ad2ant1 1130 |
. . 3
⊢ ((𝐴 ∈
No ∧ 𝐵 ∈
No ∧ 𝐴 ≠ 𝐵) → dom 𝐴 ∈ On) |
47 | | ontri1 6402 |
. . 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)})) |
48 | 44, 46, 47 | syl2anc 582 |
. 2
⊢ ((𝐴 ∈
No ∧ 𝐵 ∈
No ∧ 𝐴 ≠ 𝐵) → (∩
{𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)} ⊆ dom 𝐴 ↔ ¬ dom 𝐴 ∈ ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)})) |
49 | 43, 48 | mpbird 256 |
1
⊢ ((𝐴 ∈
No ∧ 𝐵 ∈
No ∧ 𝐴 ≠ 𝐵) → ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)} ⊆ dom 𝐴) |