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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 11338). 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 5595 | . 2 wff 𝑅 Or 𝐴 |
4 | 1, 2 | wpo 5594 | . . 3 wff 𝑅 Po 𝐴 |
5 | vx | . . . . . . . 8 setvar 𝑥 | |
6 | 5 | cv 1535 | . . . . . . 7 class 𝑥 |
7 | vy | . . . . . . . 8 setvar 𝑦 | |
8 | 7 | cv 1535 | . . . . . . 7 class 𝑦 |
9 | 6, 8, 2 | wbr 5147 | . . . . . 6 wff 𝑥𝑅𝑦 |
10 | 5, 7 | weq 1959 | . . . . . 6 wff 𝑥 = 𝑦 |
11 | 8, 6, 2 | wbr 5147 | . . . . . 6 wff 𝑦𝑅𝑥 |
12 | 9, 10, 11 | w3o 1085 | . . . . 5 wff (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥) |
13 | 12, 7, 1 | wral 3058 | . . . 4 wff ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥) |
14 | 13, 5, 1 | wral 3058 | . . 3 wff ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥) |
15 | 4, 14 | wa 395 | . 2 wff (𝑅 Po 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥)) |
16 | 3, 15 | wb 206 | 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 5603 sopo 5615 soss 5616 soeq1 5617 solin 5622 issod 5630 so0 5633 soinxp 5769 sosn 5774 cnvso 6309 isosolem 7366 sorpss 7746 dfwe2 7792 epweon 7793 soxp 8152 soseq 8182 sornom 10314 zorn2lem6 10538 tosso 18476 dfso3 35699 dfso2 35734 |
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