Theorem swopo 4354

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 2588	. . . 4 |- ( ph -> A. y e. A A. z e. A ( y R z -> -. z R y ) )
5		breq1 4048	. . . . . 6 |- ( y = x -> ( y R z <-> x R z ) )
6		breq2 4049	. . . . . . 7 |- ( y = x -> ( z R y <-> z R x ) )
7	6	notbid 669	. . . . . 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 4049	. . . . . 6 |- ( z = x -> ( x R z <-> x R x ) )
10		breq1 4048	. . . . . . 7 |- ( z = x -> ( z R x <-> x R x ) )
11	10	notbid 669	. . . . . 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 2891	. . . 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 1159	. . 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 731	. . . . 5 |- ( ( ( ph /\ ( x e. A /\ y e. A /\ z e. A ) ) /\ x R y ) -> ( z R y \/ x R z ) )
20	19	ord 726	. . . 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 4352	1 |- ( ph -> R Po A )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   \/ wo 710   /\ w3a 981   e. wcel 2176   A.wral 2484   class class class wbr 4045   Po wpo 4342

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-v 2774  df-un 3170  df-sn 3639  df-pr 3640  df-op 3642  df-br 4046  df-po 4344

This theorem is referenced by:  swoer  6650