![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 5563 | . 2 ⊢ 𝑅 Po ∅ | |
2 | ral0 4471 | . 2 ⊢ ∀𝑥 ∈ ∅ ∀𝑦 ∈ ∅ (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥) | |
3 | df-so 5547 | . 2 ⊢ (𝑅 Or ∅ ↔ (𝑅 Po ∅ ∧ ∀𝑥 ∈ ∅ ∀𝑦 ∈ ∅ (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥))) | |
4 | 1, 2, 3 | mpbir2an 710 | 1 ⊢ 𝑅 Or ∅ |
Colors of variables: wff setvar class |
Syntax hints: ∨ w3o 1087 ∀wral 3065 ∅c0 4283 class class class wbr 5106 Po wpo 5544 Or wor 5545 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-9 2117 ax-ext 2708 |
This theorem depends on definitions: df-bi 206 df-an 398 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2715 df-cleq 2729 df-ral 3066 df-dif 3914 df-nul 4284 df-po 5546 df-so 5547 |
This theorem is referenced by: we0 5629 wemapso2 9490 |
Copyright terms: Public domain | W3C validator |