Proof of Theorem nosep1o
| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | simpr 484 | . . 3
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No  ∧ 𝐴 ≠ 𝐵) ∧ (𝐴‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) = 1o) → (𝐴‘∩ {𝑥
∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) = 1o) | 
| 2 |  | nosepne 27725 | . . . . . . . . . . . 12
⊢ ((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No  ∧ 𝐴 ≠ 𝐵) → (𝐴‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) ≠ (𝐵‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)})) | 
| 3 | 2 | adantr 480 | . . . . . . . . . . 11
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No  ∧ 𝐴 ≠ 𝐵) ∧ (𝐴‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) = 1o) → (𝐴‘∩ {𝑥
∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) ≠ (𝐵‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)})) | 
| 4 | 1, 3 | eqnetrrd 3009 | . . . . . . . . . 10
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No  ∧ 𝐴 ≠ 𝐵) ∧ (𝐴‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) = 1o) → 1o
≠ (𝐵‘∩ {𝑥
∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)})) | 
| 5 | 4 | necomd 2996 | . . . . . . . . 9
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No  ∧ 𝐴 ≠ 𝐵) ∧ (𝐴‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) = 1o) → (𝐵‘∩ {𝑥
∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) ≠ 1o) | 
| 6 | 5 | neneqd 2945 | . . . . . . . 8
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No  ∧ 𝐴 ≠ 𝐵) ∧ (𝐴‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) = 1o) → ¬ (𝐵‘∩ {𝑥
∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) = 1o) | 
| 7 |  | simpl2 1193 | . . . . . . . . 9
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No  ∧ 𝐴 ≠ 𝐵) ∧ (𝐴‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) = 1o) → 𝐵 ∈  No
) | 
| 8 |  | nofv 27702 | . . . . . . . . 9
⊢ (𝐵 ∈ 
No  → ((𝐵‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) = ∅ ∨ (𝐵‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) = 1o ∨ (𝐵‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) = 2o)) | 
| 9 | 7, 8 | syl 17 | . . . . . . . 8
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No  ∧ 𝐴 ≠ 𝐵) ∧ (𝐴‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) = 1o) → ((𝐵‘∩ {𝑥
∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) = ∅ ∨ (𝐵‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) = 1o ∨ (𝐵‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) = 2o)) | 
| 10 |  | 3orel2 1486 | . . . . . . . 8
⊢ (¬
(𝐵‘∩ {𝑥
∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) = 1o → (((𝐵‘∩ {𝑥
∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) = ∅ ∨ (𝐵‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) = 1o ∨ (𝐵‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) = 2o) → ((𝐵‘∩ {𝑥
∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) = ∅ ∨ (𝐵‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) = 2o))) | 
| 11 | 6, 9, 10 | sylc 65 | . . . . . . 7
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No  ∧ 𝐴 ≠ 𝐵) ∧ (𝐴‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) = 1o) → ((𝐵‘∩ {𝑥
∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) = ∅ ∨ (𝐵‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) = 2o)) | 
| 12 |  | eqid 2737 | . . . . . . 7
⊢
1o = 1o | 
| 13 | 11, 12 | jctil 519 | . . . . . 6
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No  ∧ 𝐴 ≠ 𝐵) ∧ (𝐴‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) = 1o) → (1o =
1o ∧ ((𝐵‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) = ∅ ∨ (𝐵‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) = 2o))) | 
| 14 |  | andi 1010 | . . . . . 6
⊢
((1o = 1o ∧ ((𝐵‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) = ∅ ∨ (𝐵‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) = 2o)) ↔ ((1o
= 1o ∧ (𝐵‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) = ∅) ∨ (1o =
1o ∧ (𝐵‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) = 2o))) | 
| 15 | 13, 14 | sylib 218 | . . . . 5
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No  ∧ 𝐴 ≠ 𝐵) ∧ (𝐴‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) = 1o) → ((1o =
1o ∧ (𝐵‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) = ∅) ∨ (1o =
1o ∧ (𝐵‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) = 2o))) | 
| 16 |  | 3mix1 1331 | . . . . . 6
⊢
((1o = 1o ∧ (𝐵‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) = ∅) → ((1o =
1o ∧ (𝐵‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) = ∅) ∨ (1o =
1o ∧ (𝐵‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) = 2o) ∨ (1o =
∅ ∧ (𝐵‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) = 2o))) | 
| 17 |  | 3mix2 1332 | . . . . . 6
⊢
((1o = 1o ∧ (𝐵‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) = 2o) → ((1o =
1o ∧ (𝐵‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) = ∅) ∨ (1o =
1o ∧ (𝐵‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) = 2o) ∨ (1o =
∅ ∧ (𝐵‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) = 2o))) | 
| 18 | 16, 17 | jaoi 858 | . . . . 5
⊢
(((1o = 1o ∧ (𝐵‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) = ∅) ∨ (1o =
1o ∧ (𝐵‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) = 2o)) → ((1o
= 1o ∧ (𝐵‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) = ∅) ∨ (1o =
1o ∧ (𝐵‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) = 2o) ∨ (1o =
∅ ∧ (𝐵‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) = 2o))) | 
| 19 | 15, 18 | syl 17 | . . . 4
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No  ∧ 𝐴 ≠ 𝐵) ∧ (𝐴‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) = 1o) → ((1o =
1o ∧ (𝐵‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) = ∅) ∨ (1o =
1o ∧ (𝐵‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) = 2o) ∨ (1o =
∅ ∧ (𝐵‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) = 2o))) | 
| 20 |  | 1oex 8516 | . . . . 5
⊢
1o ∈ V | 
| 21 |  | fvex 6919 | . . . . 5
⊢ (𝐵‘∩ {𝑥
∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) ∈ V | 
| 22 | 20, 21 | brtp 5528 | . . . 4
⊢
(1o{〈1o, ∅〉, 〈1o,
2o〉, 〈∅, 2o〉} (𝐵‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) ↔ ((1o = 1o
∧ (𝐵‘∩ {𝑥
∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) = ∅) ∨ (1o =
1o ∧ (𝐵‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) = 2o) ∨ (1o =
∅ ∧ (𝐵‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) = 2o))) | 
| 23 | 19, 22 | sylibr 234 | . . 3
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No  ∧ 𝐴 ≠ 𝐵) ∧ (𝐴‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) = 1o) →
1o{〈1o, ∅〉, 〈1o,
2o〉, 〈∅, 2o〉} (𝐵‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)})) | 
| 24 | 1, 23 | eqbrtrd 5165 | . 2
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No  ∧ 𝐴 ≠ 𝐵) ∧ (𝐴‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) = 1o) → (𝐴‘∩ {𝑥
∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}){〈1o, ∅〉,
〈1o, 2o〉, 〈∅, 2o〉}
(𝐵‘∩ {𝑥
∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)})) | 
| 25 |  | simpl1 1192 | . . 3
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No  ∧ 𝐴 ≠ 𝐵) ∧ (𝐴‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) = 1o) → 𝐴 ∈  No
) | 
| 26 |  | sltval2 27701 | . . 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)}))) | 
| 27 | 25, 7, 26 | syl2anc 584 | . 2
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No  ∧ 𝐴 ≠ 𝐵) ∧ (𝐴‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) = 1o) → (𝐴 <s 𝐵 ↔ (𝐴‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}){〈1o, ∅〉,
〈1o, 2o〉, 〈∅, 2o〉}
(𝐵‘∩ {𝑥
∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}))) | 
| 28 | 24, 27 | mpbird 257 | 1
⊢ (((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No  ∧ 𝐴 ≠ 𝐵) ∧ (𝐴‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) = 1o) → 𝐴 <s 𝐵) |