Proof of Theorem nosep1o
Step | Hyp | Ref
| Expression |
1 | | simpr 477 |
. . 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 31831 |
. . . . . . . . . . . 12
⊢ ((𝐴 ∈
No ∧ 𝐵 ∈
No ∧ 𝐴 ≠ 𝐵) → (𝐴‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) ≠ (𝐵‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)})) |
3 | 2 | adantr 481 |
. . . . . . . . . . 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 2862 |
. . . . . . . . . 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 2849 |
. . . . . . . . 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 2799 |
. . . . . . . 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 1065 |
. . . . . . . . 9
⊢ (((𝐴 ∈
No ∧ 𝐵 ∈
No ∧ 𝐴 ≠ 𝐵) ∧ (𝐴‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) = 1𝑜) → 𝐵 ∈
No ) |
8 | | nofv 31810 |
. . . . . . . . 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 31592 |
. . . . . . . 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 2622 |
. . . . . . 7
⊢
1𝑜 = 1𝑜 |
13 | 11, 12 | jctil 560 |
. . . . . 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 911 |
. . . . . 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 208 |
. . . . 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 1230 |
. . . . . 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 1231 |
. . . . . 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 394 |
. . . . 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 | | 1on 7567 |
. . . . . 6
⊢
1𝑜 ∈ On |
21 | 20 | elexi 3213 |
. . . . 5
⊢
1𝑜 ∈ V |
22 | | fvex 6201 |
. . . . 5
⊢ (𝐵‘∩ {𝑥
∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) ∈ V |
23 | 21, 22 | brtp 31639 |
. . . 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))) |
24 | 19, 23 | sylibr 224 |
. . 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)})) |
25 | 1, 24 | eqbrtrd 4675 |
. 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)})) |
26 | | simpl1 1064 |
. . 3
⊢ (((𝐴 ∈
No ∧ 𝐵 ∈
No ∧ 𝐴 ≠ 𝐵) ∧ (𝐴‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) = 1𝑜) → 𝐴 ∈
No ) |
27 | | sltval2 31809 |
. . 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)}))) |
28 | 26, 7, 27 | syl2anc 693 |
. 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)}))) |
29 | 25, 28 | mpbird 247 |
1
⊢ (((𝐴 ∈
No ∧ 𝐵 ∈
No ∧ 𝐴 ≠ 𝐵) ∧ (𝐴‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) = 1𝑜) → 𝐴 <s 𝐵) |