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 10118). 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 5034 | . 2 wff 𝑅 Or 𝐴 |
4 | 1, 2 | wpo 5033 | . . 3 wff 𝑅 Po 𝐴 |
5 | vx | . . . . . . . 8 setvar 𝑥 | |
6 | 5 | cv 1482 | . . . . . . 7 class 𝑥 |
7 | vy | . . . . . . . 8 setvar 𝑦 | |
8 | 7 | cv 1482 | . . . . . . 7 class 𝑦 |
9 | 6, 8, 2 | wbr 4653 | . . . . . 6 wff 𝑥𝑅𝑦 |
10 | 5, 7 | weq 1874 | . . . . . 6 wff 𝑥 = 𝑦 |
11 | 8, 6, 2 | wbr 4653 | . . . . . 6 wff 𝑦𝑅𝑥 |
12 | 9, 10, 11 | w3o 1036 | . . . . 5 wff (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥) |
13 | 12, 7, 1 | wral 2912 | . . . 4 wff ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥) |
14 | 13, 5, 1 | wral 2912 | . . 3 wff ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥) |
15 | 4, 14 | wa 384 | . 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 5041 sopo 5052 soss 5053 soeq1 5054 solin 5058 issod 5065 so0 5068 soinxp 5183 sosn 5188 cnvso 5674 isosolem 6597 sorpss 6942 dfwe2 6981 soxp 7290 sornom 9099 zorn2lem6 9323 tosso 17036 dfso3 31601 dfso2 31644 soseq 31751 |
Copyright terms: Public domain | W3C validator |