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Mirrors > Home > MPE Home > Th. List > so0 | Structured version Visualization version GIF version |
Description: Any relation is a strict ordering of the empty set. (Contributed by NM, 16-Mar-1997.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) |
Ref | Expression |
---|---|
so0 | ⊢ 𝑅 Or ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | po0 5596 | . 2 ⊢ 𝑅 Po ∅ | |
2 | ral0 4505 | . 2 ⊢ ∀𝑥 ∈ ∅ ∀𝑦 ∈ ∅ (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥) | |
3 | df-so 5580 | . 2 ⊢ (𝑅 Or ∅ ↔ (𝑅 Po ∅ ∧ ∀𝑥 ∈ ∅ ∀𝑦 ∈ ∅ (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥))) | |
4 | 1, 2, 3 | mpbir2an 708 | 1 ⊢ 𝑅 Or ∅ |
Colors of variables: wff setvar class |
Syntax hints: ∨ w3o 1083 ∀wral 3053 ∅c0 4315 class class class wbr 5139 Po wpo 5577 Or wor 5578 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-9 2108 ax-ext 2695 |
This theorem depends on definitions: df-bi 206 df-an 396 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2702 df-cleq 2716 df-ral 3054 df-dif 3944 df-nul 4316 df-po 5579 df-so 5580 |
This theorem is referenced by: we0 5662 wemapso2 9545 |
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