![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > df-so | Structured version Visualization version GIF version |
Description: Define the strict complete (linear) order predicate. The expression 𝑅 Or 𝐴 is true if relationship 𝑅 orders 𝐴. For example, < Or ℝ is true (ltso 10302). Equivalent to Definition 6.19(1) of [TakeutiZaring] p. 29. (Contributed by NM, 21-Jan-1996.) |
Ref | Expression |
---|---|
df-so | ⊢ (𝑅 Or 𝐴 ↔ (𝑅 Po 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cA | . . 3 class 𝐴 | |
2 | cR | . . 3 class 𝑅 | |
3 | 1, 2 | wor 5178 | . 2 wff 𝑅 Or 𝐴 |
4 | 1, 2 | wpo 5177 | . . 3 wff 𝑅 Po 𝐴 |
5 | vx | . . . . . . . 8 setvar 𝑥 | |
6 | 5 | cv 1623 | . . . . . . 7 class 𝑥 |
7 | vy | . . . . . . . 8 setvar 𝑦 | |
8 | 7 | cv 1623 | . . . . . . 7 class 𝑦 |
9 | 6, 8, 2 | wbr 4796 | . . . . . 6 wff 𝑥𝑅𝑦 |
10 | 5, 7 | weq 2032 | . . . . . 6 wff 𝑥 = 𝑦 |
11 | 8, 6, 2 | wbr 4796 | . . . . . 6 wff 𝑦𝑅𝑥 |
12 | 9, 10, 11 | w3o 1071 | . . . . 5 wff (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥) |
13 | 12, 7, 1 | wral 3042 | . . . 4 wff ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥) |
14 | 13, 5, 1 | wral 3042 | . . 3 wff ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥) |
15 | 4, 14 | wa 383 | . 2 wff (𝑅 Po 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥)) |
16 | 3, 15 | wb 196 | 1 wff (𝑅 Or 𝐴 ↔ (𝑅 Po 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥))) |
Colors of variables: wff setvar class |
This definition is referenced by: nfso 5185 sopo 5196 soss 5197 soeq1 5198 solin 5202 issod 5209 so0 5212 soinxp 5332 sosn 5337 cnvso 5827 isosolem 6752 sorpss 7099 dfwe2 7138 soxp 7450 sornom 9283 zorn2lem6 9507 tosso 17229 dfso3 31900 dfso2 31943 soseq 32052 |
Copyright terms: Public domain | W3C validator |