Theorem so0 5587

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 5566	. 2 ⊢ 𝑅 Po ∅
2		ral0 4479	. 2 ⊢ ∀𝑥 ∈ ∅ ∀𝑦 ∈ ∅ (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥)
3		df-so 5550	. 2 ⊢ (𝑅 Or ∅ ↔ (𝑅 Po ∅ ∧ ∀𝑥 ∈ ∅ ∀𝑦 ∈ ∅ (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥)))
4	1, 2, 3	mpbir2an 711	1 ⊢ 𝑅 Or ∅

Syntax hints:   ∨ w3o 1085  ∀wral 3045  ∅c0 4299   class class class wbr 5110   Po wpo 5547   Or wor 5548

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 2008  ax-9 2119  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-ral 3046  df-dif 3920  df-nul 4300  df-po 5549  df-so 5550

This theorem is referenced by:  we0  5636  wemapso2  9513