Theorem swopo 4034

Description: A strict weak order is a partial order. (Contributed by Mario Carneiro, 9-Jul-2014.)

Hypotheses

Ref	Expression
swopo.1	|- ((ph /\ (y e. A /\ z e. A)) -> (y Rz -> -. z Ry))
swopo.2	|- ((ph /\ (x e. A /\ y e. A /\ z e. A)) -> (x Ry -> (x Rz \/ z Ry)))

Assertion

Ref	Expression
swopo	|- (ph -> R Po A)

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

Proof of Theorem swopo

Step	Hyp	Ref	Expression
1		id 19	. . . . 5 |- (x e. A -> x e. A)
2	1	ancli 306	. . . 4 |- (x e. A -> (x e. A /\ x e. A))
3		swopo.1	. . . . 5 |- ((ph /\ (y e. A /\ z e. A)) -> (y Rz -> -. z Ry))
4	3	ralrimivva 2395	. . . 4 |- (ph -> A.y e. A A.z e. A (y Rz -> -. z Ry))
5		breq1 3758	. . . . . 6 |- (y = x -> (y Rz <-> x Rz))
6		breq2 3759	. . . . . . 7 |- (y = x -> (z Ry <-> z Rx))
7	6	notbid 591	. . . . . 6 |- (y = x -> (-. z Ry <-> -. z Rx))
8	5, 7	imbi12d 223	. . . . 5 |- (y = x -> ((y Rz -> -. z Ry) <-> (x Rz -> -. z Rx)))
9		breq2 3759	. . . . . 6 |- (z = x -> (x Rz <-> x Rx))
10		breq1 3758	. . . . . . 7 |- (z = x -> (z Rx <-> x Rx))
11	10	notbid 591	. . . . . 6 |- (z = x -> (-. z Rx <-> -. x Rx))
12	9, 11	imbi12d 223	. . . . 5 |- (z = x -> ((x Rz -> -. z Rx) <-> (x Rx -> -. x Rx)))
13	8, 12	rspc2va 2657	. . . 4 |- (((x e. A /\ x e. A) /\ A.y e. A A.z e. A (y Rz -> -. z Ry)) -> (x Rx -> -. x Rx))
14	2, 4, 13	syl2anr 274	. . 3 |- ((ph /\ x e. A) -> (x Rx -> -. x Rx))
15	14	pm2.01d 548	. 2 |- ((ph /\ x e. A) -> -. x Rx)
16	3	3adantr1 1062	. . 3 |- ((ph /\ (x e. A /\ y e. A /\ z e. A)) -> (y Rz -> -. z Ry))
17		swopo.2	. . . . . . 7 |- ((ph /\ (x e. A /\ y e. A /\ z e. A)) -> (x Ry -> (x Rz \/ z Ry)))
18	17	imp 115	. . . . . 6 |- (((ph /\ (x e. A /\ y e. A /\ z e. A)) /\ x Ry) -> (x Rz \/ z Ry))
19	18	orcomd 647	. . . . 5 |- (((ph /\ (x e. A /\ y e. A /\ z e. A)) /\ x Ry) -> (z Ry \/ x Rz))
20	19	ord 642	. . . 4 |- (((ph /\ (x e. A /\ y e. A /\ z e. A)) /\ x Ry) -> (-. z Ry -> x Rz))
21	20	expimpd 345	. . 3 |- ((ph /\ (x e. A /\ y e. A /\ z e. A)) -> ((x Ry /\ -. z Ry) -> x Rz))
22	16, 21	sylan2d 278	. 2 |- ((ph /\ (x e. A /\ y e. A /\ z e. A)) -> ((x Ry /\ y Rz) -> x Rz))
23	15, 22	ispod 4032	1 |- (ph -> R Po A)

Syntax hints:  -. wn 3   -> wi 4   /\ wa 97   \/ wo 628   /\ w3a 884   e. wcel 1390  A.wral 2300   class class class wbr 3755   Po wpo 4022

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-v 2553  df-un 2916  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-po 4024

This theorem is referenced by:  swoer  6070