Theorem poirr2 4867

Description: A partial order relation is irreflexive. (Contributed by Mario Carneiro, 2-Nov-2015.)

Assertion

Ref	Expression
poirr2	|- ( R Po A -> ( R i^i ( _I |` A ) ) = (/) )

Proof of Theorem poirr2

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		relres 4783	. . . 4 |- Rel ( _I |` A )
2		relin2 4596	. . . 4 |- ( Rel ( _I |` A ) -> Rel ( R i^i ( _I |` A ) ) )
3	1, 2	mp1i 10	. . 3 |- ( R Po A -> Rel ( R i^i ( _I |` A ) ) )
4		df-br 3876	. . . . 5 |- ( x ( R i^i ( _I |` A ) ) y <-> <. x , y >. e. ( R i^i ( _I |` A ) ) )
5		brin 3922	. . . . 5 |- ( x ( R i^i ( _I |` A ) ) y <-> ( x R y /\ x ( _I |` A ) y ) )
6	4, 5	bitr3i 185	. . . 4 |- ( <. x , y >. e. ( R i^i ( _I |` A ) ) <-> ( x R y /\ x ( _I |` A ) y ) )
7		vex 2644	. . . . . . . . 9 |- y e. _V
8	7	brres 4761	. . . . . . . 8 |- ( x ( _I |` A ) y <-> ( x _I y /\ x e. A ) )
9		poirr 4167	. . . . . . . . . . 11 |- ( ( R Po A /\ x e. A ) -> -. x R x )
10	7	ideq 4629	. . . . . . . . . . . . 13 |- ( x _I y <-> x = y )
11		breq2 3879	. . . . . . . . . . . . 13 |- ( x = y -> ( x R x <-> x R y ) )
12	10, 11	sylbi 120	. . . . . . . . . . . 12 |- ( x _I y -> ( x R x <-> x R y ) )
13	12	notbid 633	. . . . . . . . . . 11 |- ( x _I y -> ( -. x R x <-> -. x R y ) )
14	9, 13	syl5ibcom 154	. . . . . . . . . 10 |- ( ( R Po A /\ x e. A ) -> ( x _I y -> -. x R y ) )
15	14	expimpd 358	. . . . . . . . 9 |- ( R Po A -> ( ( x e. A /\ x _I y ) -> -. x R y ) )
16	15	ancomsd 267	. . . . . . . 8 |- ( R Po A -> ( ( x _I y /\ x e. A ) -> -. x R y ) )
17	8, 16	syl5bi 151	. . . . . . 7 |- ( R Po A -> ( x ( _I |` A ) y -> -. x R y ) )
18	17	con2d 594	. . . . . 6 |- ( R Po A -> ( x R y -> -. x ( _I |` A ) y ) )
19		imnan 665	. . . . . 6 |- ( ( x R y -> -. x ( _I |` A ) y ) <-> -. ( x R y /\ x ( _I |` A ) y ) )
20	18, 19	sylib 121	. . . . 5 |- ( R Po A -> -. ( x R y /\ x ( _I |` A ) y ) )
21	20	pm2.21d 589	. . . 4 |- ( R Po A -> ( ( x R y /\ x ( _I |` A ) y ) -> <. x , y >. e. (/) ) )
22	6, 21	syl5bi 151	. . 3 |- ( R Po A -> ( <. x , y >. e. ( R i^i ( _I |` A ) ) -> <. x , y >. e. (/) ) )
23	3, 22	relssdv 4569	. 2 |- ( R Po A -> ( R i^i ( _I |` A ) ) C_ (/) )
24		ss0 3350	. 2 |- ( ( R i^i ( _I |` A ) ) C_ (/) -> ( R i^i ( _I |` A ) ) = (/) )
25	23, 24	syl 14	1 |- ( R Po A -> ( R i^i ( _I |` A ) ) = (/) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1299   e. wcel 1448   i^i cin 3020   C_ wss 3021   (/)c0 3310   <.cop 3477   class class class wbr 3875   _I cid 4148   Po wpo 4154   |` cres 4479   Rel wrel 4482

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-pow 4038  ax-pr 4069

This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ral 2380  df-rex 2381  df-v 2643  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-nul 3311  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-br 3876  df-opab 3930  df-id 4153  df-po 4156  df-xp 4483  df-rel 4484  df-res 4489

This theorem is referenced by: (None)