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Theorem ispod 5178
Description: Sufficient conditions for a partial order. (Contributed by NM, 9-Jul-2014.)
Hypotheses
Ref Expression
ispod.1 ((𝜑𝑥𝐴) → ¬ 𝑥𝑅𝑥)
ispod.2 ((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑧𝐴)) → ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
Assertion
Ref Expression
ispod (𝜑𝑅 Po 𝐴)
Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝑥,𝑅,𝑦,𝑧   𝜑,𝑥,𝑦,𝑧

Proof of Theorem ispod
StepHypRef Expression
1 ispod.1 . . . . 5 ((𝜑𝑥𝐴) → ¬ 𝑥𝑅𝑥)
213ad2antr1 1203 . . . 4 ((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑧𝐴)) → ¬ 𝑥𝑅𝑥)
3 ispod.2 . . . 4 ((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑧𝐴)) → ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
42, 3jca 501 . . 3 ((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑧𝐴)) → (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
54ralrimivvva 3121 . 2 (𝜑 → ∀𝑥𝐴𝑦𝐴𝑧𝐴𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
6 df-po 5170 . 2 (𝑅 Po 𝐴 ↔ ∀𝑥𝐴𝑦𝐴𝑧𝐴𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
75, 6sylibr 224 1 (𝜑𝑅 Po 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 382  w3a 1071  wcel 2145  wral 3061   class class class wbr 4786   Po wpo 5168
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-3an 1073  df-ral 3066  df-po 5170
This theorem is referenced by:  swopo  5180  pofun  5186  issoi  5201  wemappo  8610  pospo  17181  legso  25715  pocnv  31991
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