Theorem issod 5065

Description: An irreflexive, transitive, linear relation is a strict ordering. (Contributed by NM, 21-Jan-1996.) (Revised by Mario Carneiro, 9-Jul-2014.)

Hypotheses

Ref	Expression
issod.1	|- ( ph -> R Po A )
issod.2	|- ( ( ph /\ ( x e. A /\ y e. A ) ) -> ( x R y \/ x = y \/ y R x ) )

Assertion

Ref	Expression
issod	|- ( ph -> R Or A )

Distinct variable groups:   x, y, R   x, A, y   ph, x, y

Proof of Theorem issod

Step	Hyp	Ref	Expression
1		issod.1	. 2 |- ( ph -> R Po A )
2		issod.2	. . 3 |- ( ( ph /\ ( x e. A /\ y e. A ) ) -> ( x R y \/ x = y \/ y R x ) )
3	2	ralrimivva 2971	. 2 |- ( ph -> A. x e. A A. y e. A ( x R y \/ x = y \/ y R x ) )
4		df-so 5036	. 2 |- ( R Or A <-> ( R Po A /\ A. x e. A A. y e. A ( x R y \/ x = y \/ y R x ) ) )
5	1, 3, 4	sylanbrc 698	1 |- ( ph -> R Or A )

Syntax hints:   -> wi 4   /\ wa 384   \/ w3o 1036   e. wcel 1990   A.wral 2912   class class class wbr 4653   Po wpo 5033   Or wor 5034

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839

This theorem depends on definitions:  df-bi 197  df-an 386  df-ral 2917  df-so 5036

This theorem is referenced by:  issoi  5066  swoso  7775  wemapsolem  8455  legso  25494  fin2so  33396