MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  poirr2 Structured version   Unicode version

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.
StepHypRef Expression
1 relres 5213 . . . 4  |-  Rel  (  _I  |`  A )
2 relin2 5033 . . . 4  |-  ( Rel  (  _I  |`  A )  ->  Rel  ( R  i^i  (  _I  |`  A ) ) )
31, 2mp1i 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 ) )
64, 5bitr3i 251 . . . 4  |-  ( <.
x ,  y >.  e.  ( R  i^i  (  _I  |`  A ) )  <-> 
( x R y  /\  x (  _I  |`  A ) y ) )
7 vex 3037 . . . . . . . . 9  |-  y  e. 
_V
87brres 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 )
107ideq 5068 . . . . . . . . . . . . 13  |-  ( x  _I  y  <->  x  =  y )
11 breq2 4371 . . . . . . . . . . . . 13  |-  ( x  =  y  ->  (
x R x  <->  x R
y ) )
1210, 11sylbi 195 . . . . . . . . . . . 12  |-  ( x  _I  y  ->  (
x R x  <->  x R
y ) )
1312notbid 292 . . . . . . . . . . 11  |-  ( x  _I  y  ->  ( -.  x R x  <->  -.  x R y ) )
149, 13syl5ibcom 220 . . . . . . . . . 10  |-  ( ( R  Po  A  /\  x  e.  A )  ->  ( x  _I  y  ->  -.  x R y ) )
1514expimpd 601 . . . . . . . . 9  |-  ( R  Po  A  ->  (
( x  e.  A  /\  x  _I  y
)  ->  -.  x R y ) )
1615ancomsd 452 . . . . . . . 8  |-  ( R  Po  A  ->  (
( x  _I  y  /\  x  e.  A
)  ->  -.  x R y ) )
178, 16syl5bi 217 . . . . . . 7  |-  ( R  Po  A  ->  (
x (  _I  |`  A ) y  ->  -.  x R y ) )
1817con2d 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 ) )
2018, 19sylib 196 . . . . 5  |-  ( R  Po  A  ->  -.  ( x R y  /\  x (  _I  |`  A ) y ) )
2120pm2.21d 106 . . . 4  |-  ( R  Po  A  ->  (
( x R y  /\  x (  _I  |`  A ) y )  ->  <. x ,  y
>.  e.  (/) ) )
226, 21syl5bi 217 . . 3  |-  ( R  Po  A  ->  ( <. x ,  y >.  e.  ( R  i^i  (  _I  |`  A ) )  ->  <. x ,  y
>.  e.  (/) ) )
233, 22relssdv 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 ) )  =  (/) )
2523, 24syl 16 1  |-  ( R  Po  A  ->  ( R  i^i  (  _I  |`  A ) )  =  (/) )
Colors of variables: wff setvar class
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)
  Copyright terms: Public domain W3C validator