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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 5483 | . 2 ⊢ 𝑅 Po ∅ | |
2 | ral0 4454 | . 2 ⊢ ∀𝑥 ∈ ∅ ∀𝑦 ∈ ∅ (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥) | |
3 | df-so 5468 | . 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 1081 ∀wral 3136 ∅c0 4289 class class class wbr 5057 Po wpo 5465 Or wor 5466 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-ext 2791 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1775 df-sb 2064 df-clab 2798 df-cleq 2812 df-clel 2891 df-ral 3141 df-dif 3937 df-nul 4290 df-po 5467 df-so 5468 |
This theorem is referenced by: we0 5543 wemapso2 9009 |
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