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Theorem ispod 4322
Description: Sufficient conditions for a partial order. (Contributed by NM, 9-Jul-2014.)
Hypotheses
Ref Expression
ispod.1  |-  ( (
ph  /\  x  e.  A )  ->  -.  x R x )
ispod.2  |-  ( (
ph  /\  ( x  e.  A  /\  y  e.  A  /\  z  e.  A ) )  -> 
( ( x R y  /\  y R z )  ->  x R z ) )
Assertion
Ref Expression
ispod  |-  ( ph  ->  R  Po  A )
Distinct variable groups:    x, y,
z, A    x, R, y, z    ph, x, y, z

Proof of Theorem ispod
StepHypRef Expression
1 ispod.1 . . . . 5  |-  ( (
ph  /\  x  e.  A )  ->  -.  x R x )
213ad2antr1 1120 . . . 4  |-  ( (
ph  /\  ( x  e.  A  /\  y  e.  A  /\  z  e.  A ) )  ->  -.  x R x )
3 ispod.2 . . . 4  |-  ( (
ph  /\  ( x  e.  A  /\  y  e.  A  /\  z  e.  A ) )  -> 
( ( x R y  /\  y R z )  ->  x R z ) )
42, 3jca 518 . . 3  |-  ( (
ph  /\  ( x  e.  A  /\  y  e.  A  /\  z  e.  A ) )  -> 
( -.  x R x  /\  ( ( x R y  /\  y R z )  ->  x R z ) ) )
54ralrimivvva 2636 . 2  |-  ( ph  ->  A. x  e.  A  A. y  e.  A  A. z  e.  A  ( -.  x R x  /\  ( ( x R y  /\  y R z )  ->  x R z ) ) )
6 df-po 4314 . 2  |-  ( R  Po  A  <->  A. x  e.  A  A. y  e.  A  A. z  e.  A  ( -.  x R x  /\  (
( x R y  /\  y R z )  ->  x R
z ) ) )
75, 6sylibr 203 1  |-  ( ph  ->  R  Po  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    /\ w3a 934    e. wcel 1684   A.wral 2543   class class class wbr 4023    Po wpo 4312
This theorem is referenced by:  swopo  4324  pofun  4330  issoi  4345  wemappo  7264  pospo  14107
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-11 1715
This theorem depends on definitions:  df-bi 177  df-an 360  df-3an 936  df-nf 1532  df-ral 2548  df-po 4314
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