Theorem so0 2153

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 2138	. . 3 ⊢ (R Or ∅ ↔ (R Po ∅ ∧ ∀x ∈ ∅ ∀y ∈ ∅ (xRy ∨ x = y ∨ yRx)))
2		po0 2137	. . 3 ⊢ R Po ∅
3	1, 2	mpbiran 547	. 2 ⊢ (R Or ∅ ↔ ∀x ∈ ∅ ∀y ∈ ∅ (xRy ∨ x = y ∨ yRx))
4		noel 1711	. . 3 ⊢ ¬ x ∈ ∅
5	4	pm2.21i 73	. 2 ⊢ (x ∈ ∅ → ∀y ∈ ∅ (xRy ∨ x = y ∨ yRx))
6	3, 5	mprgbir 1250	1 ⊢ R Or ∅

Syntax hints:   ∨ w3o 580   = weq 797   ∈ wcel 1092  ∀wral 1201  ∅c0 1707   class class class wbr 2054   Po wpo 2058   Or wor 2059

This theorem is referenced by:  we0 2196

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-v 1349  df-dif 1489  df-nul 1708  df-po 2128  df-so 2138

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