Theorem swopo 4324

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 R z -> -. z R y ) )
swopo.2	|- ( ( ph /\ ( x e. A /\ y e. A /\ z e. A ) ) -> ( x R y -> ( x R z \/ z R y ) ) )

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 323	. . . 4 |- ( x e. A -> ( x e. A /\ x e. A ) )
3		swopo.1	. . . . 5 |- ( ( ph /\ ( y e. A /\ z e. A ) ) -> ( y R z -> -. z R y ) )
4	3	ralrimivva 2572	. . . 4 |- ( ph -> A. y e. A A. z e. A ( y R z -> -. z R y ) )
5		breq1 4021	. . . . . 6 |- ( y = x -> ( y R z <-> x R z ) )
6		breq2 4022	. . . . . . 7 |- ( y = x -> ( z R y <-> z R x ) )
7	6	notbid 668	. . . . . 6 |- ( y = x -> ( -. z R y <-> -. z R x ) )
8	5, 7	imbi12d 234	. . . . 5 |- ( y = x -> ( ( y R z -> -. z R y ) <-> ( x R z -> -. z R x ) ) )
9		breq2 4022	. . . . . 6 |- ( z = x -> ( x R z <-> x R x ) )
10		breq1 4021	. . . . . . 7 |- ( z = x -> ( z R x <-> x R x ) )
11	10	notbid 668	. . . . . 6 |- ( z = x -> ( -. z R x <-> -. x R x ) )
12	9, 11	imbi12d 234	. . . . 5 |- ( z = x -> ( ( x R z -> -. z R x ) <-> ( x R x -> -. x R x ) ) )
13	8, 12	rspc2va 2870	. . . 4 |- ( ( ( x e. A /\ x e. A ) /\ A. y e. A A. z e. A ( y R z -> -. z R y ) ) -> ( x R x -> -. x R x ) )
14	2, 4, 13	syl2anr 290	. . 3 |- ( ( ph /\ x e. A ) -> ( x R x -> -. x R x ) )
15	14	pm2.01d 619	. 2 |- ( ( ph /\ x e. A ) -> -. x R x )
16	3	3adantr1 1158	. . 3 |- ( ( ph /\ ( x e. A /\ y e. A /\ z e. A ) ) -> ( y R z -> -. z R y ) )
17		swopo.2	. . . . . . 7 |- ( ( ph /\ ( x e. A /\ y e. A /\ z e. A ) ) -> ( x R y -> ( x R z \/ z R y ) ) )
18	17	imp 124	. . . . . 6 |- ( ( ( ph /\ ( x e. A /\ y e. A /\ z e. A ) ) /\ x R y ) -> ( x R z \/ z R y ) )
19	18	orcomd 730	. . . . 5 |- ( ( ( ph /\ ( x e. A /\ y e. A /\ z e. A ) ) /\ x R y ) -> ( z R y \/ x R z ) )
20	19	ord 725	. . . 4 |- ( ( ( ph /\ ( x e. A /\ y e. A /\ z e. A ) ) /\ x R y ) -> ( -. z R y -> x R z ) )
21	20	expimpd 363	. . 3 |- ( ( ph /\ ( x e. A /\ y e. A /\ z e. A ) ) -> ( ( x R y /\ -. z R y ) -> x R z ) )
22	16, 21	sylan2d 294	. 2 |- ( ( ph /\ ( x e. A /\ y e. A /\ z e. A ) ) -> ( ( x R y /\ y R z ) -> x R z ) )
23	15, 22	ispod 4322	1 |- ( ph -> R Po A )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   \/ wo 709   /\ w3a 980   e. wcel 2160   A.wral 2468   class class class wbr 4018   Po wpo 4312

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-v 2754  df-un 3148  df-sn 3613  df-pr 3614  df-op 3616  df-br 4019  df-po 4314

This theorem is referenced by:  swoer  6586