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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 5459 | . 2 ⊢ 𝑅 Po ∅ | |
2 | ral0 4399 | . 2 ⊢ ∀𝑥 ∈ ∅ ∀𝑦 ∈ ∅ (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥) | |
3 | df-so 5443 | . 2 ⊢ (𝑅 Or ∅ ↔ (𝑅 Po ∅ ∧ ∀𝑥 ∈ ∅ ∀𝑦 ∈ ∅ (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥))) | |
4 | 1, 2, 3 | mpbir2an 711 | 1 ⊢ 𝑅 Or ∅ |
Colors of variables: wff setvar class |
Syntax hints: ∨ w3o 1087 ∀wral 3053 ∅c0 4211 class class class wbr 5030 Po wpo 5440 Or wor 5441 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-9 2124 ax-ext 2710 |
This theorem depends on definitions: df-bi 210 df-an 400 df-tru 1545 df-fal 1555 df-ex 1787 df-sb 2075 df-clab 2717 df-cleq 2730 df-ral 3058 df-dif 3846 df-nul 4212 df-po 5442 df-so 5443 |
This theorem is referenced by: we0 5520 wemapso2 9090 |
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