Theorem swopo 4397

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 2612	. . . 4 |- ( ph -> A. y e. A A. z e. A ( y R z -> -. z R y ) )
5		breq1 4086	. . . . . 6 |- ( y = x -> ( y R z <-> x R z ) )
6		breq2 4087	. . . . . . 7 |- ( y = x -> ( z R y <-> z R x ) )
7	6	notbid 671	. . . . . 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 4087	. . . . . 6 |- ( z = x -> ( x R z <-> x R x ) )
10		breq1 4086	. . . . . . 7 |- ( z = x -> ( z R x <-> x R x ) )
11	10	notbid 671	. . . . . 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 2921	. . . 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 621	. 2 |- ( ( ph /\ x e. A ) -> -. x R x )
16	3	3adantr1 1180	. . 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 734	. . . . 5 |- ( ( ( ph /\ ( x e. A /\ y e. A /\ z e. A ) ) /\ x R y ) -> ( z R y \/ x R z ) )
20	19	ord 729	. . . 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 4395	1 |- ( ph -> R Po A )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   \/ wo 713   /\ w3a 1002   e. wcel 2200   A.wral 2508   class class class wbr 4083   Po wpo 4385

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-v 2801  df-un 3201  df-sn 3672  df-pr 3673  df-op 3675  df-br 4084  df-po 4387

This theorem is referenced by:  swoer  6708