Theorem so0 5598

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 5577	. 2 ⊢ 𝑅 Po ∅
2		ral0 4455	. 2 ⊢ ∀𝑥 ∈ ∅ ∀𝑦 ∈ ∅ (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥)
3		df-so 5561	. 2 ⊢ (𝑅 Or ∅ ↔ (𝑅 Po ∅ ∧ ∀𝑥 ∈ ∅ ∀𝑦 ∈ ∅ (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥)))
4	1, 2, 3	mpbir2an 723	1 ⊢ 𝑅 Or ∅

Syntax hints:   ∨ w3o 1100  ∀wral 3079  ∅c0 4288   class class class wbr 5105   Po wpo 5558   Or wor 5559

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-9 2155  ax-ext 2737

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-ral 3080  df-dif 3910  df-nul 4289  df-po 5560  df-so 5561

This theorem is referenced by:  we0  5647  wemapso2  9503