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Theorem ispod 4032
Description: Sufficient conditions for a partial order. (Contributed by NM, 9-Jul-2014.)
Hypotheses
Ref Expression
ispod.1 ((φ x A) → ¬ x𝑅x)
ispod.2 ((φ (x A y A z A)) → ((x𝑅y y𝑅z) → 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 ((φ x A) → ¬ x𝑅x)
213ad2antr1 1068 . . . 4 ((φ (x A y A z A)) → ¬ x𝑅x)
3 ispod.2 . . . 4 ((φ (x A y A z A)) → ((x𝑅y y𝑅z) → x𝑅z))
42, 3jca 290 . . 3 ((φ (x A y A z A)) → (¬ x𝑅x ((x𝑅y y𝑅z) → x𝑅z)))
54ralrimivvva 2396 . 2 (φx A y A z Ax𝑅x ((x𝑅y y𝑅z) → x𝑅z)))
6 df-po 4024 . 2 (𝑅 Po Ax A y A z Ax𝑅x ((x𝑅y y𝑅z) → x𝑅z)))
75, 6sylibr 137 1 (φ𝑅 Po A)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   wa 97   w3a 884   wcel 1390  wral 2300   class class class wbr 3755   Po wpo 4022
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 1333  ax-gen 1335  ax-4 1397  ax-17 1416
This theorem depends on definitions:  df-bi 110  df-3an 886  df-nf 1347  df-ral 2305  df-po 4024
This theorem is referenced by:  swopo  4034  pofun  4040  ltsopi  6304  ltsonq  6382  ltpopr  6569  ltposr  6691  ltso  6893  xrltso  8487
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