Theorem poirr2 5389

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 5299	. . . 4 |- Rel ( _I |` A )
2		relin2 5119	. . . 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 4448	. . . . 5 |- ( x ( R i^i ( _I |` A ) ) y <-> <. x , y >. e. ( R i^i ( _I |` A ) ) )
5		brin 4496	. . . . 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 3116	. . . . . . . . 9 |- y e. _V
8	7	brres 5278	. . . . . . . 8 |- ( x ( _I |` A ) y <-> ( x _I y /\ x e. A ) )
9		poirr 4811	. . . . . . . . . . 11 |- ( ( R Po A /\ x e. A ) -> -. x R x )
10	7	ideq 5153	. . . . . . . . . . . . 13 |- ( x _I y <-> x = y )
11		breq2 4451	. . . . . . . . . . . . 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 294	. . . . . . . . . . 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 603	. . . . . . . . 9 |- ( R Po A -> ( ( x e. A /\ x _I y ) -> -. x R y ) )
16	15	ancomsd 454	. . . . . . . 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 422	. . . . . 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 5093	. 2 |- ( R Po A -> ( R i^i ( _I |` A ) ) C_ (/) )
24		ss0 3816	. 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 369   = wceq 1379   e. wcel 1767   i^i cin 3475   C_ wss 3476   (/)c0 3785   <.cop 4033   class class class wbr 4447   _I cid 4790   Po wpo 4798   |` cres 5001   Rel wrel 5004

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-br 4448  df-opab 4506  df-id 4795  df-po 4800  df-xp 5005  df-rel 5006  df-res 5011

This theorem is referenced by: (None)