Theorem issoi 4537

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 454	. . . 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 454	. . . 4 |- ( ( T. /\ ( x e. A /\ y e. A /\ z e. A ) ) -> ( ( x R y /\ y R z ) -> x R z ) )
5	2, 4	ispod 4514	. . 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 454	. . 3 |- ( ( T. /\ ( x e. A /\ y e. A ) ) -> ( x R y \/ x = y \/ y R x ) )
8	5, 7	issod 4536	. 2 |- ( T. -> R Or A )
9	8	trud 1333	1 |- R Or A

Syntax hints:   -. wn 3   -> wi 4   /\ wa 360   \/ w3o 936   /\ w3a 937   T. wtru 1326   e. wcel 1726   class class class wbr 4215   Or wor 4505

This theorem is referenced by:  isso2i  4538  ltsopr  8914  sltsolem1  25628

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-11 1762

This theorem depends on definitions:  df-bi 179  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-ral 2712  df-po 4506  df-so 4507