Proof of Theorem nosepnelem
| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | sltval2 27702 | . . 3
⊢ ((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) → (𝐴 <s 𝐵 ↔ (𝐴‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}){〈1o, ∅〉,
〈1o, 2o〉, 〈∅, 2o〉}
(𝐵‘∩ {𝑥
∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}))) | 
| 2 |  | fvex 6918 | . . . . 5
⊢ (𝐴‘∩ {𝑥
∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) ∈ V | 
| 3 |  | fvex 6918 | . . . . 5
⊢ (𝐵‘∩ {𝑥
∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) ∈ V | 
| 4 | 2, 3 | brtp 5527 | . . . 4
⊢ ((𝐴‘∩ {𝑥
∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}){〈1o, ∅〉,
〈1o, 2o〉, 〈∅, 2o〉}
(𝐵‘∩ {𝑥
∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) ↔ (((𝐴‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) = 1o ∧ (𝐵‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) = ∅) ∨ ((𝐴‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) = 1o ∧ (𝐵‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) = 2o) ∨ ((𝐴‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) = ∅ ∧ (𝐵‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) = 2o))) | 
| 5 |  | 1n0 8527 | . . . . . 6
⊢
1o ≠ ∅ | 
| 6 |  | simpl 482 | . . . . . . 7
⊢ (((𝐴‘∩ {𝑥
∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) = 1o ∧ (𝐵‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) = ∅) → (𝐴‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) = 1o) | 
| 7 |  | simpr 484 | . . . . . . 7
⊢ (((𝐴‘∩ {𝑥
∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) = 1o ∧ (𝐵‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) = ∅) → (𝐵‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) = ∅) | 
| 8 | 6, 7 | neeq12d 3001 | . . . . . 6
⊢ (((𝐴‘∩ {𝑥
∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) = 1o ∧ (𝐵‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) = ∅) → ((𝐴‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) ≠ (𝐵‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) ↔ 1o ≠
∅)) | 
| 9 | 5, 8 | mpbiri 258 | . . . . 5
⊢ (((𝐴‘∩ {𝑥
∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) = 1o ∧ (𝐵‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) = ∅) → (𝐴‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) ≠ (𝐵‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)})) | 
| 10 |  | df-2o 8508 | . . . . . . . . . . 11
⊢
2o = suc 1o | 
| 11 |  | df-1o 8507 | . . . . . . . . . . 11
⊢
1o = suc ∅ | 
| 12 | 10, 11 | eqeq12i 2754 | . . . . . . . . . 10
⊢
(2o = 1o ↔ suc 1o = suc
∅) | 
| 13 |  | 1on 8519 | . . . . . . . . . . 11
⊢
1o ∈ On | 
| 14 |  | 0elon 6437 | . . . . . . . . . . 11
⊢ ∅
∈ On | 
| 15 |  | suc11 6490 | . . . . . . . . . . 11
⊢
((1o ∈ On ∧ ∅ ∈ On) → (suc
1o = suc ∅ ↔ 1o = ∅)) | 
| 16 | 13, 14, 15 | mp2an 692 | . . . . . . . . . 10
⊢ (suc
1o = suc ∅ ↔ 1o = ∅) | 
| 17 | 12, 16 | bitri 275 | . . . . . . . . 9
⊢
(2o = 1o ↔ 1o =
∅) | 
| 18 | 17 | necon3bii 2992 | . . . . . . . 8
⊢
(2o ≠ 1o ↔ 1o ≠
∅) | 
| 19 | 5, 18 | mpbir 231 | . . . . . . 7
⊢
2o ≠ 1o | 
| 20 | 19 | necomi 2994 | . . . . . 6
⊢
1o ≠ 2o | 
| 21 |  | simpl 482 | . . . . . . 7
⊢ (((𝐴‘∩ {𝑥
∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) = 1o ∧ (𝐵‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) = 2o) → (𝐴‘∩ {𝑥
∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) = 1o) | 
| 22 |  | simpr 484 | . . . . . . 7
⊢ (((𝐴‘∩ {𝑥
∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) = 1o ∧ (𝐵‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) = 2o) → (𝐵‘∩ {𝑥
∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) = 2o) | 
| 23 | 21, 22 | neeq12d 3001 | . . . . . 6
⊢ (((𝐴‘∩ {𝑥
∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) = 1o ∧ (𝐵‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) = 2o) → ((𝐴‘∩ {𝑥
∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) ≠ (𝐵‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) ↔ 1o ≠
2o)) | 
| 24 | 20, 23 | mpbiri 258 | . . . . 5
⊢ (((𝐴‘∩ {𝑥
∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) = 1o ∧ (𝐵‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) = 2o) → (𝐴‘∩ {𝑥
∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) ≠ (𝐵‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)})) | 
| 25 |  | 2on 8521 | . . . . . . . . 9
⊢
2o ∈ On | 
| 26 | 25 | elexi 3502 | . . . . . . . 8
⊢
2o ∈ V | 
| 27 | 26 | prid2 4762 | . . . . . . 7
⊢
2o ∈ {1o, 2o} | 
| 28 | 27 | nosgnn0i 27705 | . . . . . 6
⊢ ∅
≠ 2o | 
| 29 |  | simpl 482 | . . . . . . 7
⊢ (((𝐴‘∩ {𝑥
∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) = ∅ ∧ (𝐵‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) = 2o) → (𝐴‘∩ {𝑥
∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) = ∅) | 
| 30 |  | simpr 484 | . . . . . . 7
⊢ (((𝐴‘∩ {𝑥
∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) = ∅ ∧ (𝐵‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) = 2o) → (𝐵‘∩ {𝑥
∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) = 2o) | 
| 31 | 29, 30 | neeq12d 3001 | . . . . . 6
⊢ (((𝐴‘∩ {𝑥
∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) = ∅ ∧ (𝐵‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) = 2o) → ((𝐴‘∩ {𝑥
∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) ≠ (𝐵‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) ↔ ∅ ≠
2o)) | 
| 32 | 28, 31 | mpbiri 258 | . . . . 5
⊢ (((𝐴‘∩ {𝑥
∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) = ∅ ∧ (𝐵‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) = 2o) → (𝐴‘∩ {𝑥
∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) ≠ (𝐵‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)})) | 
| 33 | 9, 24, 32 | 3jaoi 1429 | . . . 4
⊢ ((((𝐴‘∩ {𝑥
∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) = 1o ∧ (𝐵‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) = ∅) ∨ ((𝐴‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) = 1o ∧ (𝐵‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) = 2o) ∨ ((𝐴‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) = ∅ ∧ (𝐵‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) = 2o)) → (𝐴‘∩ {𝑥
∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) ≠ (𝐵‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)})) | 
| 34 | 4, 33 | sylbi 217 | . . 3
⊢ ((𝐴‘∩ {𝑥
∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}){〈1o, ∅〉,
〈1o, 2o〉, 〈∅, 2o〉}
(𝐵‘∩ {𝑥
∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) → (𝐴‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) ≠ (𝐵‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)})) | 
| 35 | 1, 34 | biimtrdi 253 | . 2
⊢ ((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) → (𝐴 <s 𝐵 → (𝐴‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) ≠ (𝐵‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}))) | 
| 36 | 35 | 3impia 1117 | 1
⊢ ((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No  ∧ 𝐴 <s 𝐵) → (𝐴‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) ≠ (𝐵‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)})) |