Theorem itlso 2919

Description: A irreflexive, transitive, linear relation is a strict ordering.

Hypotheses

Ref	Expression
itlso.1	|- (x e. A -> -. xRx)
itlso.2	|- ((x e. A /\ y e. A /\ z e. A) -> ((xRy /\ yRz) -> xRz))
itlso.3	|- ((x e. A /\ y e. A) -> (xRy \/ x = y \/ yRx))

Assertion

Ref	Expression
itlso	|- R Or A

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

Proof of Theorem itlso

Step	Hyp	Ref	Expression
1		df-so 2906	. 2 |- (R Or A <-> (R Po A /\ A.x e. A A.y e. A (xRy \/ x = y \/ yRx)))
2		itlso.1	. . . . . 6 |- (x e. A -> -. xRx)
3	2	3ad2ant1 812	. . . . 5 |- ((x e. A /\ y e. A /\ z e. A) -> -. xRx)
4		itlso.2	. . . . 5 |- ((x e. A /\ y e. A /\ z e. A) -> ((xRy /\ yRz) -> xRz))
5	3, 4	jca 295	. . . 4 |- ((x e. A /\ y e. A /\ z e. A) -> (-. xRx /\ ((xRy /\ yRz) -> xRz)))
6	5	rgen3 1771	. . 3 |- A.x e. A A.y e. A A.z e. A (-. xRx /\ ((xRy /\ yRz) -> xRz))
7		df-po 2896	. . 3 |- (R Po A <-> A.x e. A A.y e. A A.z e. A (-. xRx /\ ((xRy /\ yRz) -> xRz)))
8	6, 7	mpbir 197	. 2 |- R Po A
9		itlso.3	. . 3 |- ((x e. A /\ y e. A) -> (xRy \/ x = y \/ yRx))
10	9	rgen2a 1746	. 2 |- A.x e. A A.y e. A (xRy \/ x = y \/ yRx)
11	1, 8, 10	mpbir2an 742	1 |- R Or A

Syntax hints:  -. wn 2   -> wi 3   /\ wa 230   \/ w3o 786   /\ w3a 787   = wceq 997   e. wcel 999  A.wral 1692   class class class wbr 2674   Po wpo 2894   Or wor 2895

This theorem is referenced by:  so 2920  ltsopr 5201

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1003  ax-gen 1004  ax-8 1005  ax-10 1007  ax-12 1009  ax-17 1012  ax-4 1014  ax-5o 1016  ax-6o 1019  ax-9o 1164  ax-10o 1182  ax-11o 1260  ax-ext 1504

This theorem depends on definitions:  df-bi 154  df-an 232  df-3an 789  df-ex 1022  df-sb 1214  df-cleq 1515  df-clel 1518  df-ral 1696  df-po 2896  df-so 2906

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