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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 5492 | . 2 ⊢ 𝑅 Po ∅ | |
2 | ral0 4458 | . 2 ⊢ ∀𝑥 ∈ ∅ ∀𝑦 ∈ ∅ (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥) | |
3 | df-so 5477 | . 2 ⊢ (𝑅 Or ∅ ↔ (𝑅 Po ∅ ∧ ∀𝑥 ∈ ∅ ∀𝑦 ∈ ∅ (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥))) | |
4 | 1, 2, 3 | mpbir2an 709 | 1 ⊢ 𝑅 Or ∅ |
Colors of variables: wff setvar class |
Syntax hints: ∨ w3o 1082 ∀wral 3140 ∅c0 4293 class class class wbr 5068 Po wpo 5474 Or wor 5475 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1781 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-ral 3145 df-dif 3941 df-nul 4294 df-po 5476 df-so 5477 |
This theorem is referenced by: we0 5552 wemapso2 9019 |
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