Theorem ispod 4267

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 1147	. . . 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 304	. . 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 2540	. 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 4259	. 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 133	1 |- ( ph -> R Po A )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   /\ w3a 963   e. wcel 2128   A.wral 2435   class class class wbr 3967   Po wpo 4257

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1427  ax-gen 1429  ax-4 1490  ax-17 1506

This theorem depends on definitions:  df-bi 116  df-3an 965  df-nf 1441  df-ral 2440  df-po 4259

This theorem is referenced by:  swopo  4269  pofun  4275  wepo  4322  ltsopi  7243  ltsonq  7321  ltpopr  7518  ltposr  7686  ltso  7958  xrltso  9710