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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 4513 | . 2 ⊢ ∀𝑥 ∈ ∅ ∀𝑦 ∈ ∅ ∀𝑧 ∈ ∅ (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)) | |
| 2 | df-po 5592 | . 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 3061 ∅c0 4333 class class class wbr 5143 Po wpo 5590 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-ral 3062 df-dif 3954 df-nul 4334 df-po 5592 |
| This theorem is referenced by: so0 5630 posn 5771 dfpo2 6316 ipo0 44468 |
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