MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  issod Structured version   Visualization version   GIF version

Theorem issod 5201
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 3120 . 2 (𝜑 → ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥))
4 df-so 5172 . 2 (𝑅 Or 𝐴 ↔ (𝑅 Po 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)))
51, 3, 4sylanbrc 572 1 (𝜑𝑅 Or 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  w3o 1070  wcel 2145  wral 3061   class class class wbr 4787   Po wpo 5169   Or wor 5170
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991
This theorem depends on definitions:  df-bi 197  df-an 383  df-ral 3066  df-so 5172
This theorem is referenced by:  issoi  5202  swoso  7933  wemapsolem  8615  legso  25715  fin2so  33729
  Copyright terms: Public domain W3C validator