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Theorem ispod 4122
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 1108 . . . 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 300 . . 3  |-  ( (
ph  /\  ( x  e.  A  /\  y  e.  A  /\  z  e.  A ) )  -> 
( -.  x R x  /\  ( ( x R y  /\  y R z )  ->  x R z ) ) )
54ralrimivvva 2456 . 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 4114 . 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 132 1  |-  ( ph  ->  R  Po  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    /\ w3a 924    e. wcel 1438   A.wral 2359   class class class wbr 3837    Po wpo 4112
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-gen 1383  ax-4 1445  ax-17 1464
This theorem depends on definitions:  df-bi 115  df-3an 926  df-nf 1395  df-ral 2364  df-po 4114
This theorem is referenced by:  swopo  4124  pofun  4130  wepo  4177  ltsopi  6858  ltsonq  6936  ltpopr  7133  ltposr  7288  ltso  7542  xrltso  9235
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