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Theorem ispod 5512
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 1187 . . . 4 ((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑧𝐴)) → ¬ 𝑥𝑅𝑥)
3 ispod.2 . . . 4 ((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑧𝐴)) → ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
42, 3jca 512 . . 3 ((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑧𝐴)) → (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
54ralrimivvva 3127 . 2 (𝜑 → ∀𝑥𝐴𝑦𝐴𝑧𝐴𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
6 df-po 5503 . 2 (𝑅 Po 𝐴 ↔ ∀𝑥𝐴𝑦𝐴𝑧𝐴𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
75, 6sylibr 233 1 (𝜑𝑅 Po 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396  w3a 1086  wcel 2106  wral 3064   class class class wbr 5074   Po wpo 5501
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913
This theorem depends on definitions:  df-bi 206  df-an 397  df-3an 1088  df-ral 3069  df-po 5503
This theorem is referenced by:  swopo  5514  pofun  5521  issoi  5537  wemappo  9308  pospo  18063  legso  26960  pocnv  33730  poxp2  33790  poxp3  33796  lrrecpo  34098
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