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Theorem ispod 4037
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 1069 . . . 4 ((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑧𝐴)) → ¬ 𝑥𝑅𝑥)
3 ispod.2 . . . 4 ((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑧𝐴)) → ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
42, 3jca 290 . . 3 ((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑧𝐴)) → (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
54ralrimivvva 2399 . 2 (𝜑 → ∀𝑥𝐴𝑦𝐴𝑧𝐴𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
6 df-po 4029 . 2 (𝑅 Po 𝐴 ↔ ∀𝑥𝐴𝑦𝐴𝑧𝐴𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
75, 6sylibr 137 1 (𝜑𝑅 Po 𝐴)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 97  w3a 885  wcel 1393  wral 2303   class class class wbr 3760   Po wpo 4027
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-4 1400  ax-17 1419
This theorem depends on definitions:  df-bi 110  df-3an 887  df-nf 1350  df-ral 2308  df-po 4029
This theorem is referenced by:  swopo  4039  pofun  4045  ltsopi  6361  ltsonq  6439  ltpopr  6636  ltposr  6791  ltso  7038  xrltso  8646
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