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| 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 5609 | . 2 ⊢ 𝑅 Po ∅ | |
| 2 | ral0 4513 | . 2 ⊢ ∀𝑥 ∈ ∅ ∀𝑦 ∈ ∅ (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥) | |
| 3 | df-so 5593 | . 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 1086 ∀wral 3061 ∅c0 4333 class class class wbr 5143 Po wpo 5590 Or wor 5591 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-9 2118 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-ral 3062 df-dif 3954 df-nul 4334 df-po 5592 df-so 5593 | 
| This theorem is referenced by: we0 5680 wemapso2 9593 | 
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