Theorem so0 5309

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 5290	. 2 ⊢ 𝑅 Po ∅
2		ral0 4299	. 2 ⊢ ∀𝑥 ∈ ∅ ∀𝑦 ∈ ∅ (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥)
3		df-so 5275	. 2 ⊢ (𝑅 Or ∅ ↔ (𝑅 Po ∅ ∧ ∀𝑥 ∈ ∅ ∀𝑦 ∈ ∅ (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥)))
4	1, 2, 3	mpbir2an 701	1 ⊢ 𝑅 Or ∅

Syntax hints:   ∨ w3o 1070  ∀wral 3090  ∅c0 4141   class class class wbr 4886   Po wpo 5272   Or wor 5273

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-ext 2754

This theorem depends on definitions:  df-bi 199  df-an 387  df-ex 1824  df-sb 2012  df-clab 2764  df-cleq 2770  df-clel 2774  df-ral 3095  df-dif 3795  df-nul 4142  df-po 5274  df-so 5275

This theorem is referenced by:  we0  5350  wemapso2  8747