Theorem swopo 4776

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 23	. . . . 5 |- ( x e. A -> x e. A )
2	1	ancli 553	. . . 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 2844	. . . 4 |- ( ph -> A. y e. A A. z e. A ( y R z -> -. z R y ) )
5		breq1 4420	. . . . . 6 |- ( y = x -> ( y R z <-> x R z ) )
6		breq2 4421	. . . . . . 7 |- ( y = x -> ( z R y <-> z R x ) )
7	6	notbid 295	. . . . . 6 |- ( y = x -> ( -. z R y <-> -. z R x ) )
8	5, 7	imbi12d 321	. . . . 5 |- ( y = x -> ( ( y R z -> -. z R y ) <-> ( x R z -> -. z R x ) ) )
9		breq2 4421	. . . . . 6 |- ( z = x -> ( x R z <-> x R x ) )
10		breq1 4420	. . . . . . 7 |- ( z = x -> ( z R x <-> x R x ) )
11	10	notbid 295	. . . . . 6 |- ( z = x -> ( -. z R x <-> -. x R x ) )
12	9, 11	imbi12d 321	. . . . 5 |- ( z = x -> ( ( x R z -> -. z R x ) <-> ( x R x -> -. x R x ) ) )
13	8, 12	rspc2va 3189	. . . 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 480	. . 3 |- ( ( ph /\ x e. A ) -> ( x R x -> -. x R x ) )
15	14	pm2.01d 172	. 2 |- ( ( ph /\ x e. A ) -> -. x R x )
16	3	3adantr1 1164	. . 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 430	. . . . . 6 |- ( ( ( ph /\ ( x e. A /\ y e. A /\ z e. A ) ) /\ x R y ) -> ( x R z \/ z R y ) )
19	18	orcomd 389	. . . . 5 |- ( ( ( ph /\ ( x e. A /\ y e. A /\ z e. A ) ) /\ x R y ) -> ( z R y \/ x R z ) )
20	19	ord 378	. . . 4 |- ( ( ( ph /\ ( x e. A /\ y e. A /\ z e. A ) ) /\ x R y ) -> ( -. z R y -> x R z ) )
21	20	expimpd 606	. . 3 |- ( ( ph /\ ( x e. A /\ y e. A /\ z e. A ) ) -> ( ( x R y /\ -. z R y ) -> x R z ) )
22	16, 21	sylan2d 484	. 2 |- ( ( ph /\ ( x e. A /\ y e. A /\ z e. A ) ) -> ( ( x R y /\ y R z ) -> x R z ) )
23	15, 22	ispod 4774	1 |- ( ph -> R Po A )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 369   /\ wa 370   /\ w3a 982   e. wcel 1867   A.wral 2773   class class class wbr 4417   Po wpo 4764

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ral 2778  df-rab 2782  df-v 3080  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-sn 3994  df-pr 3996  df-op 4000  df-br 4418  df-po 4766

This theorem is referenced by:  swoer  7390  swoso  7393