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Metamath Proof Explorer
Description: An irreflexive, transitive, linear relation is a strict ordering.
(Contributed by NM, 21-Jan-1996) (Revised by Mario Carneiro, 9-Jul-2014)
|
|
Ref |
Expression |
|
Hypotheses |
issoi.1 |
⊢ ( 𝑥 ∈ 𝐴 → ¬ 𝑥 𝑅 𝑥 ) |
|
|
issoi.2 |
⊢ ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴 ) → ( ( 𝑥 𝑅 𝑦 ∧ 𝑦 𝑅 𝑧 ) → 𝑥 𝑅 𝑧 ) ) |
|
|
issoi.3 |
⊢ ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ) → ( 𝑥 𝑅 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 𝑅 𝑥 ) ) |
|
Assertion |
issoi |
⊢ 𝑅 Or 𝐴 |
Proof
| Step |
Hyp |
Ref |
Expression |
| 1 |
|
issoi.1 |
⊢ ( 𝑥 ∈ 𝐴 → ¬ 𝑥 𝑅 𝑥 ) |
| 2 |
|
issoi.2 |
⊢ ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴 ) → ( ( 𝑥 𝑅 𝑦 ∧ 𝑦 𝑅 𝑧 ) → 𝑥 𝑅 𝑧 ) ) |
| 3 |
|
issoi.3 |
⊢ ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ) → ( 𝑥 𝑅 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 𝑅 𝑥 ) ) |
| 4 |
1
|
adantl |
⊢ ( ( ⊤ ∧ 𝑥 ∈ 𝐴 ) → ¬ 𝑥 𝑅 𝑥 ) |
| 5 |
2
|
adantl |
⊢ ( ( ⊤ ∧ ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴 ) ) → ( ( 𝑥 𝑅 𝑦 ∧ 𝑦 𝑅 𝑧 ) → 𝑥 𝑅 𝑧 ) ) |
| 6 |
4 5
|
ispod |
⊢ ( ⊤ → 𝑅 Po 𝐴 ) |
| 7 |
3
|
adantl |
⊢ ( ( ⊤ ∧ ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ) ) → ( 𝑥 𝑅 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 𝑅 𝑥 ) ) |
| 8 |
6 7
|
issod |
⊢ ( ⊤ → 𝑅 Or 𝐴 ) |
| 9 |
8
|
mptru |
⊢ 𝑅 Or 𝐴 |