Proof of Theorem nosepssdm
| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | nosepne 27726 | . . . 4
⊢ ((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No  ∧ 𝐴 ≠ 𝐵) → (𝐴‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) ≠ (𝐵‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)})) | 
| 2 | 1 | neneqd 2944 | . . 3
⊢ ((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No  ∧ 𝐴 ≠ 𝐵) → ¬ (𝐴‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) = (𝐵‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)})) | 
| 3 |  | nodmord 27699 | . . . . . . . . 9
⊢ (𝐴 ∈ 
No  → Ord dom 𝐴) | 
| 4 | 3 | 3ad2ant1 1133 | . . . . . . . 8
⊢ ((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No  ∧ 𝐴 ≠ 𝐵) → Ord dom 𝐴) | 
| 5 |  | ordn2lp 6403 | . . . . . . . 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 399 | . . . . . . 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 234 | . . . . . 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 406 | . . . . 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 6940 | . . . . 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 27731 | . . . . . 6
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No  ∧ 𝐴 ≠ 𝐵) ∧ dom 𝐴 ∈ ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) → (𝐴‘dom 𝐴) = (𝐵‘dom 𝐴)) | 
| 13 |  | simpl1 1191 | . . . . . . . . . 10
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No  ∧ 𝐴 ≠ 𝐵) ∧ dom 𝐴 ∈ ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) → 𝐴 ∈  No
) | 
| 14 |  | ordirr 6401 | . . . . . . . . . 10
⊢ (Ord dom
𝐴 → ¬ dom 𝐴 ∈ dom 𝐴) | 
| 15 |  | ndmfv 6940 | . . . . . . . . . 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 2738 | . . . . . . . 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 2743 | . . . . . . . 8
⊢ (∅
= (𝐵‘dom 𝐴) ↔ (𝐵‘dom 𝐴) = ∅) | 
| 19 | 17, 18 | bitrdi 287 | . . . . . . 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 1192 | . . . . . . . . . . 11
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No  ∧ 𝐴 ≠ 𝐵) ∧ dom 𝐴 ∈ ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) → 𝐵 ∈  No
) | 
| 21 |  | nofun 27695 | . . . . . . . . . . 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 27704 | . . . . . . . . . . 11
⊢  ¬
∅ ∈ {1o, 2o} | 
| 24 |  | norn 27697 | . . . . . . . . . . . . 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 3981 | . . . . . . . . . . 11
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No  ∧ 𝐴 ≠ 𝐵) ∧ dom 𝐴 ∈ ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) → (∅ ∈ ran 𝐵 → ∅ ∈
{1o, 2o})) | 
| 27 | 23, 26 | mtoi 199 | . . . . . . . . . 10
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No  ∧ 𝐴 ≠ 𝐵) ∧ dom 𝐴 ∈ ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) → ¬ ∅ ∈ ran 𝐵) | 
| 28 |  | funeldmb 7380 | . . . . . . . . . 10
⊢ ((Fun
𝐵 ∧ ¬ ∅
∈ ran 𝐵) → (dom
𝐴 ∈ dom 𝐵 ↔ (𝐵‘dom 𝐴) ≠ ∅)) | 
| 29 | 22, 27, 28 | syl2anc 584 | . . . . . . . . 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 2983 | . . . . . . . 8
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No  ∧ 𝐴 ≠ 𝐵) ∧ dom 𝐴 ∈ ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) → ((𝐵‘dom 𝐴) = ∅ ↔ ¬ dom 𝐴 ∈ dom 𝐵)) | 
| 31 |  | nodmord 27699 | . . . . . . . . . . . 12
⊢ (𝐵 ∈ 
No  → Ord dom 𝐵) | 
| 32 | 31 | 3ad2ant2 1134 | . . . . . . . . . . 11
⊢ ((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No  ∧ 𝐴 ≠ 𝐵) → Ord dom 𝐵) | 
| 33 |  | ordtr1 6426 | . . . . . . . . . . 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 452 | . . . . . . . . 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 240 | . . . . . . 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 240 | . . . . . 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 6940 | . . . . 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 2779 | . . 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 815 | . 2
⊢ ((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No  ∧ 𝐴 ≠ 𝐵) → ¬ dom 𝐴 ∈ ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) | 
| 44 |  | nosepon 27711 | . . 3
⊢ ((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No  ∧ 𝐴 ≠ 𝐵) → ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)} ∈ On) | 
| 45 |  | nodmon 27696 | . . . 4
⊢ (𝐴 ∈ 
No  → dom 𝐴
∈ On) | 
| 46 | 45 | 3ad2ant1 1133 | . . 3
⊢ ((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No  ∧ 𝐴 ≠ 𝐵) → dom 𝐴 ∈ On) | 
| 47 |  | ontri1 6417 | . . 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 584 | . 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 257 | 1
⊢ ((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No  ∧ 𝐴 ≠ 𝐵) → ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)} ⊆ dom 𝐴) |