Theorem so0 5625

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.)

Assertion

Ref	Expression
so0	⊢ 𝑅 Or ∅

Proof of Theorem so0

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		po0 5606	. 2 ⊢ 𝑅 Po ∅
2		ral0 4513	. 2 ⊢ ∀𝑥 ∈ ∅ ∀𝑦 ∈ ∅ (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥)
3		df-so 5590	. 2 ⊢ (𝑅 Or ∅ ↔ (𝑅 Po ∅ ∧ ∀𝑥 ∈ ∅ ∀𝑦 ∈ ∅ (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥)))
4	1, 2, 3	mpbir2an 710	1 ⊢ 𝑅 Or ∅

Syntax hints:   ∨ w3o 1087  ∀wral 3062  ∅c0 4323   class class class wbr 5149   Po wpo 5587   Or wor 5588

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 2704

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 2711  df-cleq 2725  df-ral 3063  df-dif 3952  df-nul 4324  df-po 5589  df-so 5590

This theorem is referenced by:  we0  5672  wemapso2  9548