Theorem poirr2 4996

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 4912	. . . 4 |- Rel ( _I |` A )
2		relin2 4723	. . . 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 3983	. . . . 5 |- ( x ( R i^i ( _I |` A ) ) y <-> <. x , y >. e. ( R i^i ( _I |` A ) ) )
5		brin 4034	. . . . 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 2729	. . . . . . . . 9 |- y e. _V
8	7	brres 4890	. . . . . . . 8 |- ( x ( _I |` A ) y <-> ( x _I y /\ x e. A ) )
9		poirr 4285	. . . . . . . . . . 11 |- ( ( R Po A /\ x e. A ) -> -. x R x )
10	7	ideq 4756	. . . . . . . . . . . . 13 |- ( x _I y <-> x = y )
11		breq2 3986	. . . . . . . . . . . . 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 657	. . . . . . . . . . 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 361	. . . . . . . . 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 614	. . . . . 6 |- ( R Po A -> ( x R y -> -. x ( _I |` A ) y ) )
19		imnan 680	. . . . . 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 609	. . . 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 4696	. 2 |- ( R Po A -> ( R i^i ( _I |` A ) ) C_ (/) )
24		ss0 3449	. 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 1343   e. wcel 2136   i^i cin 3115   C_ wss 3116   (/)c0 3409   <.cop 3579   class class class wbr 3982   _I cid 4266   Po wpo 4272   |` cres 4606   Rel wrel 4609

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983  df-opab 4044  df-id 4271  df-po 4274  df-xp 4610  df-rel 4611  df-res 4616

This theorem is referenced by: (None)