Theorem issoi 4494

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
issoi.1	|- ( x e. A -> -. x R x )
issoi.2	|- ( ( x e. A /\ y e. A /\ z e. A ) -> ( ( x R y /\ y R z ) -> x R z ) )
issoi.3	|- ( ( x e. A /\ y e. A ) -> ( x R y \/ x = y \/ y R x ) )

Assertion

Ref	Expression
issoi	|- R Or A

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

Proof of Theorem issoi

Step	Hyp	Ref	Expression
1		issoi.1	. . . . 5 |- ( x e. A -> -. x R x )
2	1	adantl 453	. . . 4 |- ( ( T. /\ x e. A ) -> -. x R x )
3		issoi.2	. . . . 5 |- ( ( x e. A /\ y e. A /\ z e. A ) -> ( ( x R y /\ y R z ) -> x R z ) )
4	3	adantl 453	. . . 4 |- ( ( T. /\ ( x e. A /\ y e. A /\ z e. A ) ) -> ( ( x R y /\ y R z ) -> x R z ) )
5	2, 4	ispod 4471	. . 3 |- ( T. -> R Po A )
6		issoi.3	. . . 4 |- ( ( x e. A /\ y e. A ) -> ( x R y \/ x = y \/ y R x ) )
7	6	adantl 453	. . 3 |- ( ( T. /\ ( x e. A /\ y e. A ) ) -> ( x R y \/ x = y \/ y R x ) )
8	5, 7	issod 4493	. 2 |- ( T. -> R Or A )
9	8	trud 1329	1 |- R Or A

Syntax hints:   -. wn 3   -> wi 4   /\ wa 359   \/ w3o 935   /\ w3a 936   T. wtru 1322   e. wcel 1721   class class class wbr 4172   Or wor 4462

This theorem is referenced by:  isso2i  4495  ltsopr  8865  sltsolem1  25536

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-11 1757

This theorem depends on definitions:  df-bi 178  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-ral 2671  df-po 4463  df-so 4464