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Mirrors > Home > MPE Home > Th. List > po0 | Structured version Visualization version GIF version |
Description: Any relation is a partial order on the empty set. (Contributed by NM, 28-Mar-1997.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) |
Ref | Expression |
---|---|
po0 | ⊢ 𝑅 Po ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ral0 4519 | . 2 ⊢ ∀𝑥 ∈ ∅ ∀𝑦 ∈ ∅ ∀𝑧 ∈ ∅ (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)) | |
2 | df-po 5597 | . 2 ⊢ (𝑅 Po ∅ ↔ ∀𝑥 ∈ ∅ ∀𝑦 ∈ ∅ ∀𝑧 ∈ ∅ (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧))) | |
3 | 1, 2 | mpbir 231 | 1 ⊢ 𝑅 Po ∅ |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∀wral 3059 ∅c0 4339 class class class wbr 5148 Po wpo 5595 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-ral 3060 df-dif 3966 df-nul 4340 df-po 5597 |
This theorem is referenced by: so0 5634 posn 5774 dfpo2 6318 ipo0 44445 |
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