Theorem so0 2871

Description: Any relation is a strict ordering of the empty set.

Assertion

Ref	Expression
so0	⊢ R Or ∅

Proof of Theorem so0

Step	Hyp	Ref	Expression
1		df-so 2856	. . 3 ⊢ (R Or ∅ ↔ (R Po ∅ ⋀ ∀x ∈ ∅ ∀y ∈ ∅ (xRy ⋁ x = y ⋁ yRx)))
2		po0 2855	. . 3 ⊢ R Po ∅
3	1, 2	mpbiran 730	. 2 ⊢ (R Or ∅ ↔ ∀x ∈ ∅ ∀y ∈ ∅ (xRy ⋁ x = y ⋁ yRx))
4		noel 2287	. . 3 ⊢ ¬ x ∈ ∅
5	4	pm2.21i 77	. 2 ⊢ (x ∈ ∅ → ∀y ∈ ∅ (xRy ⋁ x = y ⋁ yRx))
6	3, 5	mprgbir 1704	1 ⊢ R Or ∅

Syntax hints:   ⋁ w3o 776   = wceq 958   ∈ wcel 960  ∀wral 1648  ∅c0 2283   class class class wbr 2624   Po wpo 2844   Or wor 2845

This theorem is referenced by:  we0 2950

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-ral 1652  df-v 1815  df-dif 2052  df-nul 2284  df-po 2846  df-so 2856

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