Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > sopo | GIF version |
Description: A strict linear order is a strict partial order. (Contributed by NM, 28-Mar-1997.) |
Ref | Expression |
---|---|
sopo | ⊢ (𝑅 Or 𝐴 → 𝑅 Po 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-iso 4052 | . 2 ⊢ (𝑅 Or 𝐴 ↔ (𝑅 Po 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑥𝑅𝑦 → (𝑥𝑅𝑧 ∨ 𝑧𝑅𝑦)))) | |
2 | 1 | simplbi 268 | 1 ⊢ (𝑅 Or 𝐴 → 𝑅 Po 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∨ wo 661 ∀wral 2348 class class class wbr 3785 Po wpo 4049 Or wor 4050 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 |
This theorem depends on definitions: df-bi 115 df-iso 4052 |
This theorem is referenced by: sonr 4072 sotr 4073 so2nr 4076 so3nr 4077 sosng 4431 |
Copyright terms: Public domain | W3C validator |