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Theorem issod 5025
Description: An irreflexive, transitive, linear relation is a strict ordering. (Contributed by NM, 21-Jan-1996.) (Revised by Mario Carneiro, 9-Jul-2014.)
Hypotheses
Ref Expression
issod.1 (𝜑𝑅 Po 𝐴)
issod.2 ((𝜑 ∧ (𝑥𝐴𝑦𝐴)) → (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥))
Assertion
Ref Expression
issod (𝜑𝑅 Or 𝐴)
Distinct variable groups:   𝑥,𝑦,𝑅   𝑥,𝐴,𝑦   𝜑,𝑥,𝑦

Proof of Theorem issod
StepHypRef Expression
1 issod.1 . 2 (𝜑𝑅 Po 𝐴)
2 issod.2 . . 3 ((𝜑 ∧ (𝑥𝐴𝑦𝐴)) → (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥))
32ralrimivva 2965 . 2 (𝜑 → ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥))
4 df-so 4996 . 2 (𝑅 Or 𝐴 ↔ (𝑅 Po 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)))
51, 3, 4sylanbrc 697 1 (𝜑𝑅 Or 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3o 1035  wcel 1987  wral 2907   class class class wbr 4613   Po wpo 4993   Or wor 4994
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836
This theorem depends on definitions:  df-bi 197  df-an 386  df-ral 2912  df-so 4996
This theorem is referenced by:  issoi  5026  swoso  7720  wemapsolem  8399  legso  25394  socnv  31363  fin2so  33028
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