Theorem ispod 4444

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

Step	Hyp	Ref	Expression
1		ispod.1	. . . . 5 |- ( ( ph /\ x e. A ) -> -. x R x )
2	1	3ad2antr1 1193	. . . 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 ) )
4	2, 3	jca 306	. . 3 |- ( ( ph /\ ( x e. A /\ y e. A /\ z e. A ) ) -> ( -. x R x /\ ( ( x R y /\ y R z ) -> x R z ) ) )
5	4	ralrimivvva 2633	. 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 4436	. 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 ) ) )
7	5, 6	sylibr 134	1 |- ( ph -> R Po A )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   /\ w3a 1009   e. wcel 2209   A.wral 2528   class class class wbr 4125   Po wpo 4434

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-gen 1502  ax-4 1563  ax-17 1579

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-nf 1514  df-ral 2533  df-po 4436

This theorem is referenced by:  swopo  4446  pofun  4452  wepo  4499  ltsopi  7677  ltsonq  7755  ltpopr  7952  ltposr  8120  ltso  8393  xrltso  10177