Theorem sonr 2861

Description: A strict order relation is irreflexive.

Assertion

Ref	Expression
sonr	⊢ ((R Or A ⋀ B ∈ A) → ¬ BRB)

Proof of Theorem sonr

Step	Hyp	Ref	Expression
1		poirr 2851	. 2 ⊢ ((R Po A ⋀ B ∈ A) → ¬ BRB)
2		sopo 2857	. 2 ⊢ (R Or A → R Po A)
3	1, 2	sylan 450	1 ⊢ ((R Or A ⋀ B ∈ A) → ¬ BRB)

Syntax hints:  ¬ wn 2   → wi 3   ⋀ wa 223   ∈ wcel 960   class class class wbr 2624   Po wpo 2844   Or wor 2845

This theorem is referenced by:  sotric 2866  sotrieq 2867  soirri 3448  suppr 4599  supsnALT 4601  1ne0sr 5217  ltnrt 5542

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-3an 779  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-ral 1652  df-v 1815  df-un 2053  df-sn 2416  df-pr 2417  df-op 2420  df-br 2625  df-po 2846  df-so 2856

Copyright terms: Public domain