Theorem swopo 4650

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 22	. . . . 5 |- ( x e. A -> x e. A )
2	1	ancli 551	. . . 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 2807	. . . 4 |- ( ph -> A. y e. A A. z e. A ( y R z -> -. z R y ) )
5		breq1 4294	. . . . . 6 |- ( y = x -> ( y R z <-> x R z ) )
6		breq2 4295	. . . . . . 7 |- ( y = x -> ( z R y <-> z R x ) )
7	6	notbid 294	. . . . . 6 |- ( y = x -> ( -. z R y <-> -. z R x ) )
8	5, 7	imbi12d 320	. . . . 5 |- ( y = x -> ( ( y R z -> -. z R y ) <-> ( x R z -> -. z R x ) ) )
9		breq2 4295	. . . . . 6 |- ( z = x -> ( x R z <-> x R x ) )
10		breq1 4294	. . . . . . 7 |- ( z = x -> ( z R x <-> x R x ) )
11	10	notbid 294	. . . . . 6 |- ( z = x -> ( -. z R x <-> -. x R x ) )
12	9, 11	imbi12d 320	. . . . 5 |- ( z = x -> ( ( x R z -> -. z R x ) <-> ( x R x -> -. x R x ) ) )
13	8, 12	rspc2va 3079	. . . 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 478	. . 3 |- ( ( ph /\ x e. A ) -> ( x R x -> -. x R x ) )
15	14	pm2.01d 169	. 2 |- ( ( ph /\ x e. A ) -> -. x R x )
16	3	3adantr1 1147	. . 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 429	. . . . . 6 |- ( ( ( ph /\ ( x e. A /\ y e. A /\ z e. A ) ) /\ x R y ) -> ( x R z \/ z R y ) )
19	18	orcomd 388	. . . . 5 |- ( ( ( ph /\ ( x e. A /\ y e. A /\ z e. A ) ) /\ x R y ) -> ( z R y \/ x R z ) )
20	19	ord 377	. . . 4 |- ( ( ( ph /\ ( x e. A /\ y e. A /\ z e. A ) ) /\ x R y ) -> ( -. z R y -> x R z ) )
21	20	expimpd 603	. . 3 |- ( ( ph /\ ( x e. A /\ y e. A /\ z e. A ) ) -> ( ( x R y /\ -. z R y ) -> x R z ) )
22	16, 21	sylan2d 482	. 2 |- ( ( ph /\ ( x e. A /\ y e. A /\ z e. A ) ) -> ( ( x R y /\ y R z ) -> x R z ) )
23	15, 22	ispod 4648	1 |- ( ph -> R Po A )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 368   /\ wa 369   /\ w3a 965   e. wcel 1756   A.wral 2714   class class class wbr 4291   Po wpo 4638

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ral 2719  df-rab 2723  df-v 2973  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-nul 3637  df-if 3791  df-sn 3877  df-pr 3879  df-op 3883  df-br 4292  df-po 4640

This theorem is referenced by:  swoer  7128  swoso  7131