Theorem poirr2 4660

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 4582	. . . 4 |- Rel ( _I |` A)
2		relin2 4399	. . . 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 3756	. . . . 5 |- (x( R i^i ( _I |` A))y <-> <.x , y >. e. ( R i^i ( _I |` A)))
5		brin 3802	. . . . 5 |- (x( R i^i ( _I |` A))y <-> (x Ry /\ x( _I |` A)y))
6	4, 5	bitr3i 175	. . . 4 |- ( <.x , y >. e. ( R i^i ( _I |` A)) <-> (x Ry /\ x( _I |` A)y))
7		vex 2554	. . . . . . . . 9 |- y e. _V
8	7	brres 4561	. . . . . . . 8 |- (x( _I |` A)y <-> (x _I y /\ x e. A))
9		poirr 4035	. . . . . . . . . . 11 |- (( R Po A /\ x e. A) -> -. x Rx)
10	7	ideq 4431	. . . . . . . . . . . . 13 |- (x _I y <-> x = y)
11		breq2 3759	. . . . . . . . . . . . 13 |- (x = y -> (x Rx <-> x Ry))
12	10, 11	sylbi 114	. . . . . . . . . . . 12 |- (x _I y -> (x Rx <-> x Ry))
13	12	notbid 591	. . . . . . . . . . 11 |- (x _I y -> (-. x Rx <-> -. x Ry))
14	9, 13	syl5ibcom 144	. . . . . . . . . 10 |- (( R Po A /\ x e. A) -> (x _I y -> -. x Ry))
15	14	expimpd 345	. . . . . . . . 9 |- ( R Po A -> ((x e. A /\ x _I y) -> -. x Ry))
16	15	ancomsd 256	. . . . . . . 8 |- ( R Po A -> ((x _I y /\ x e. A) -> -. x Ry))
17	8, 16	syl5bi 141	. . . . . . 7 |- ( R Po A -> (x( _I |` A)y -> -. x Ry))
18	17	con2d 554	. . . . . 6 |- ( R Po A -> (x Ry -> -. x( _I |` A)y))
19		imnan 623	. . . . . 6 |- ((x Ry -> -. x( _I |` A)y) <-> -. (x Ry /\ x( _I |` A)y))
20	18, 19	sylib 127	. . . . 5 |- ( R Po A -> -. (x Ry /\ x( _I |` A)y))
21	20	pm2.21d 549	. . . 4 |- ( R Po A -> ((x Ry /\ x( _I |` A)y) -> <.x , y >. e. (/)))
22	6, 21	syl5bi 141	. . 3 |- ( R Po A -> ( <.x , y >. e. ( R i^i ( _I |` A)) -> <.x , y >. e. (/)))
23	3, 22	relssdv 4375	. 2 |- ( R Po A -> ( R i^i ( _I |` A)) C_ (/))
24		ss0 3251	. 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 97   <-> wb 98   = wceq 1242   e. wcel 1390   i^i cin 2910   C_ wss 2911   (/)c0 3218   <.cop 3370   class class class wbr 3755   _I cid 4016   Po wpo 4022   |` cres 4290   Rel wrel 4293

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-id 4021  df-po 4024  df-xp 4294  df-rel 4295  df-res 4300

This theorem is referenced by: (None)