Theorem po0 2137

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

Assertion

Ref	Expression
po0	⊢ R Po ∅

Proof of Theorem po0

Step	Hyp	Ref	Expression
1		df-po 2128	. 2 ⊢ (R Po ∅ ↔ ∀x ∈ ∅ ∀y ∈ ∅ ∀z ∈ ∅ (¬ xRx ∧ ((xRy ∧ yRz) → xRz)))
2		noel 1711	. . 3 ⊢ ¬ x ∈ ∅
3	2	pm2.21i 73	. 2 ⊢ (x ∈ ∅ → ∀y ∈ ∅ ∀z ∈ ∅ (¬ xRx ∧ ((xRy ∧ yRz) → xRz)))
4	1, 3	mprgbir 1250	1 ⊢ R Po ∅

Syntax hints:  ¬ wn 1   → wi 2   ∧ wa 196   ∈ wcel 1092  ∀wral 1201  ∅c0 1707   class class class wbr 2054   Po wpo 2058

This theorem is referenced by:  so0 2153

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

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