Proof of Theorem swoord1
Step | Hyp | Ref
| Expression |
1 | | id 19 |
. . . 4
⊢ (𝜑 → 𝜑) |
2 | | swoord.6 |
. . . . 5
⊢ (𝜑 → 𝐴𝑅𝐵) |
3 | | swoer.1 |
. . . . . . 7
⊢ 𝑅 = ((𝑋 × 𝑋) ∖ ( < ∪ ◡ < )) |
4 | | difss 3253 |
. . . . . . 7
⊢ ((𝑋 × 𝑋) ∖ ( < ∪ ◡ < )) ⊆ (𝑋 × 𝑋) |
5 | 3, 4 | eqsstri 3179 |
. . . . . 6
⊢ 𝑅 ⊆ (𝑋 × 𝑋) |
6 | 5 | ssbri 4033 |
. . . . 5
⊢ (𝐴𝑅𝐵 → 𝐴(𝑋 × 𝑋)𝐵) |
7 | | df-br 3990 |
. . . . . 6
⊢ (𝐴(𝑋 × 𝑋)𝐵 ↔ 〈𝐴, 𝐵〉 ∈ (𝑋 × 𝑋)) |
8 | | opelxp1 4645 |
. . . . . 6
⊢
(〈𝐴, 𝐵〉 ∈ (𝑋 × 𝑋) → 𝐴 ∈ 𝑋) |
9 | 7, 8 | sylbi 120 |
. . . . 5
⊢ (𝐴(𝑋 × 𝑋)𝐵 → 𝐴 ∈ 𝑋) |
10 | 2, 6, 9 | 3syl 17 |
. . . 4
⊢ (𝜑 → 𝐴 ∈ 𝑋) |
11 | | swoord.5 |
. . . 4
⊢ (𝜑 → 𝐶 ∈ 𝑋) |
12 | | swoord.4 |
. . . 4
⊢ (𝜑 → 𝐵 ∈ 𝑋) |
13 | | swoer.3 |
. . . . 5
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑥 < 𝑦 → (𝑥 < 𝑧 ∨ 𝑧 < 𝑦))) |
14 | 13 | swopolem 4290 |
. . . 4
⊢ ((𝜑 ∧ (𝐴 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐴 < 𝐶 → (𝐴 < 𝐵 ∨ 𝐵 < 𝐶))) |
15 | 1, 10, 11, 12, 14 | syl13anc 1235 |
. . 3
⊢ (𝜑 → (𝐴 < 𝐶 → (𝐴 < 𝐵 ∨ 𝐵 < 𝐶))) |
16 | 3 | brdifun 6540 |
. . . . . . 7
⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝑅𝐵 ↔ ¬ (𝐴 < 𝐵 ∨ 𝐵 < 𝐴))) |
17 | 10, 12, 16 | syl2anc 409 |
. . . . . 6
⊢ (𝜑 → (𝐴𝑅𝐵 ↔ ¬ (𝐴 < 𝐵 ∨ 𝐵 < 𝐴))) |
18 | 2, 17 | mpbid 146 |
. . . . 5
⊢ (𝜑 → ¬ (𝐴 < 𝐵 ∨ 𝐵 < 𝐴)) |
19 | | orc 707 |
. . . . 5
⊢ (𝐴 < 𝐵 → (𝐴 < 𝐵 ∨ 𝐵 < 𝐴)) |
20 | 18, 19 | nsyl 623 |
. . . 4
⊢ (𝜑 → ¬ 𝐴 < 𝐵) |
21 | | biorf 739 |
. . . 4
⊢ (¬
𝐴 < 𝐵 → (𝐵 < 𝐶 ↔ (𝐴 < 𝐵 ∨ 𝐵 < 𝐶))) |
22 | 20, 21 | syl 14 |
. . 3
⊢ (𝜑 → (𝐵 < 𝐶 ↔ (𝐴 < 𝐵 ∨ 𝐵 < 𝐶))) |
23 | 15, 22 | sylibrd 168 |
. 2
⊢ (𝜑 → (𝐴 < 𝐶 → 𝐵 < 𝐶)) |
24 | 13 | swopolem 4290 |
. . . 4
⊢ ((𝜑 ∧ (𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → (𝐵 < 𝐶 → (𝐵 < 𝐴 ∨ 𝐴 < 𝐶))) |
25 | 1, 12, 11, 10, 24 | syl13anc 1235 |
. . 3
⊢ (𝜑 → (𝐵 < 𝐶 → (𝐵 < 𝐴 ∨ 𝐴 < 𝐶))) |
26 | | olc 706 |
. . . . 5
⊢ (𝐵 < 𝐴 → (𝐴 < 𝐵 ∨ 𝐵 < 𝐴)) |
27 | 18, 26 | nsyl 623 |
. . . 4
⊢ (𝜑 → ¬ 𝐵 < 𝐴) |
28 | | biorf 739 |
. . . 4
⊢ (¬
𝐵 < 𝐴 → (𝐴 < 𝐶 ↔ (𝐵 < 𝐴 ∨ 𝐴 < 𝐶))) |
29 | 27, 28 | syl 14 |
. . 3
⊢ (𝜑 → (𝐴 < 𝐶 ↔ (𝐵 < 𝐴 ∨ 𝐴 < 𝐶))) |
30 | 25, 29 | sylibrd 168 |
. 2
⊢ (𝜑 → (𝐵 < 𝐶 → 𝐴 < 𝐶)) |
31 | 23, 30 | impbid 128 |
1
⊢ (𝜑 → (𝐴 < 𝐶 ↔ 𝐵 < 𝐶)) |