Theorem poirr2 5175

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 5086	. . . 4 |- Rel ( _I |` A )
2		relin2 4891	. . . 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 4126	. . . . 5 |- ( x ( R i^i ( _I |` A ) ) y <-> <. x , y >. e. ( R i^i ( _I |` A ) ) )
5		brin 4178	. . . . 5 |- ( x ( R i^i ( _I |` A ) ) y <-> ( x R y /\ x ( _I |` A ) y ) )
6	4, 5	bitr3i 186	. . . 4 |- ( <. x , y >. e. ( R i^i ( _I |` A ) ) <-> ( x R y /\ x ( _I |` A ) y ) )
7		vex 2824	. . . . . . . . 9 |- y e. _V
8	7	brres 5064	. . . . . . . 8 |- ( x ( _I |` A ) y <-> ( x _I y /\ x e. A ) )
9		poirr 4447	. . . . . . . . . . 11 |- ( ( R Po A /\ x e. A ) -> -. x R x )
10	7	ideq 4927	. . . . . . . . . . . . 13 |- ( x _I y <-> x = y )
11		breq2 4129	. . . . . . . . . . . . 13 |- ( x = y -> ( x R x <-> x R y ) )
12	10, 11	sylbi 121	. . . . . . . . . . . 12 |- ( x _I y -> ( x R x <-> x R y ) )
13	12	notbid 677	. . . . . . . . . . 11 |- ( x _I y -> ( -. x R x <-> -. x R y ) )
14	9, 13	syl5ibcom 155	. . . . . . . . . 10 |- ( ( R Po A /\ x e. A ) -> ( x _I y -> -. x R y ) )
15	14	expimpd 363	. . . . . . . . 9 |- ( R Po A -> ( ( x e. A /\ x _I y ) -> -. x R y ) )
16	15	ancomsd 269	. . . . . . . 8 |- ( R Po A -> ( ( x _I y /\ x e. A ) -> -. x R y ) )
17	8, 16	biimtrid 152	. . . . . . 7 |- ( R Po A -> ( x ( _I |` A ) y -> -. x R y ) )
18	17	con2d 633	. . . . . 6 |- ( R Po A -> ( x R y -> -. x ( _I |` A ) y ) )
19		imnan 701	. . . . . 6 |- ( ( x R y -> -. x ( _I |` A ) y ) <-> -. ( x R y /\ x ( _I |` A ) y ) )
20	18, 19	sylib 122	. . . . 5 |- ( R Po A -> -. ( x R y /\ x ( _I |` A ) y ) )
21	20	pm2.21d 628	. . . 4 |- ( R Po A -> ( ( x R y /\ x ( _I |` A ) y ) -> <. x , y >. e. (/) ) )
22	6, 21	biimtrid 152	. . 3 |- ( R Po A -> ( <. x , y >. e. ( R i^i ( _I |` A ) ) -> <. x , y >. e. (/) ) )
23	3, 22	relssdv 4862	. 2 |- ( R Po A -> ( R i^i ( _I |` A ) ) C_ (/) )
24		ss0 3563	. 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 104   <-> wb 105   = wceq 1402   e. wcel 2209   i^i cin 3219   C_ wss 3220   (/)c0 3520   <.cop 3708   class class class wbr 4125   _I cid 4428   Po wpo 4434   |` cres 4771   Rel wrel 4774

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-br 4126  df-opab 4188  df-id 4433  df-po 4436  df-xp 4775  df-rel 4776  df-res 4781

This theorem is referenced by: (None)