Theorem poirr2 5304

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 5213	. . . 4 |- Rel ( _I |` A )
2		relin2 5033	. . . 4 |- ( Rel ( _I |` A ) -> Rel ( R i^i ( _I |` A ) ) )
3	1, 2	mp1i 12	. . 3 |- ( R Po A -> Rel ( R i^i ( _I |` A ) ) )
4		df-br 4368	. . . . 5 |- ( x ( R i^i ( _I |` A ) ) y <-> <. x , y >. e. ( R i^i ( _I |` A ) ) )
5		brin 4416	. . . . 5 |- ( x ( R i^i ( _I |` A ) ) y <-> ( x R y /\ x ( _I |` A ) y ) )
6	4, 5	bitr3i 251	. . . 4 |- ( <. x , y >. e. ( R i^i ( _I |` A ) ) <-> ( x R y /\ x ( _I |` A ) y ) )
7		vex 3037	. . . . . . . . 9 |- y e. _V
8	7	brres 5192	. . . . . . . 8 |- ( x ( _I |` A ) y <-> ( x _I y /\ x e. A ) )
9		poirr 4725	. . . . . . . . . . 11 |- ( ( R Po A /\ x e. A ) -> -. x R x )
10	7	ideq 5068	. . . . . . . . . . . . 13 |- ( x _I y <-> x = y )
11		breq2 4371	. . . . . . . . . . . . 13 |- ( x = y -> ( x R x <-> x R y ) )
12	10, 11	sylbi 195	. . . . . . . . . . . 12 |- ( x _I y -> ( x R x <-> x R y ) )
13	12	notbid 292	. . . . . . . . . . 11 |- ( x _I y -> ( -. x R x <-> -. x R y ) )
14	9, 13	syl5ibcom 220	. . . . . . . . . 10 |- ( ( R Po A /\ x e. A ) -> ( x _I y -> -. x R y ) )
15	14	expimpd 601	. . . . . . . . 9 |- ( R Po A -> ( ( x e. A /\ x _I y ) -> -. x R y ) )
16	15	ancomsd 452	. . . . . . . 8 |- ( R Po A -> ( ( x _I y /\ x e. A ) -> -. x R y ) )
17	8, 16	syl5bi 217	. . . . . . 7 |- ( R Po A -> ( x ( _I |` A ) y -> -. x R y ) )
18	17	con2d 115	. . . . . 6 |- ( R Po A -> ( x R y -> -. x ( _I |` A ) y ) )
19		imnan 420	. . . . . 6 |- ( ( x R y -> -. x ( _I |` A ) y ) <-> -. ( x R y /\ x ( _I |` A ) y ) )
20	18, 19	sylib 196	. . . . 5 |- ( R Po A -> -. ( x R y /\ x ( _I |` A ) y ) )
21	20	pm2.21d 106	. . . 4 |- ( R Po A -> ( ( x R y /\ x ( _I |` A ) y ) -> <. x , y >. e. (/) ) )
22	6, 21	syl5bi 217	. . 3 |- ( R Po A -> ( <. x , y >. e. ( R i^i ( _I |` A ) ) -> <. x , y >. e. (/) ) )
23	3, 22	relssdv 5008	. 2 |- ( R Po A -> ( R i^i ( _I |` A ) ) C_ (/) )
24		ss0 3743	. 2 |- ( ( R i^i ( _I |` A ) ) C_ (/) -> ( R i^i ( _I |` A ) ) = (/) )
25	23, 24	syl 16	1 |- ( R Po A -> ( R i^i ( _I |` A ) ) = (/) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   /\ wa 367   = wceq 1399   e. wcel 1826   i^i cin 3388   C_ wss 3389   (/)c0 3711   <.cop 3950   class class class wbr 4367   _I cid 4704   Po wpo 4712   |` cres 4915   Rel wrel 4918

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pr 4601

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-rab 2741  df-v 3036  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-sn 3945  df-pr 3947  df-op 3951  df-br 4368  df-opab 4426  df-id 4709  df-po 4714  df-xp 4919  df-rel 4920  df-res 4925

This theorem is referenced by: (None)