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Theorem poirr2 4787
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 4706 . . . 4  |-  Rel  (  _I  |`  A )
2 relin2 4522 . . . 4  |-  ( Rel  (  _I  |`  A )  ->  Rel  ( R  i^i  (  _I  |`  A ) ) )
31, 2mp1i 10 . . 3  |-  ( R  Po  A  ->  Rel  ( R  i^i  (  _I  |`  A ) ) )
4 df-br 3820 . . . . 5  |-  ( x ( R  i^i  (  _I  |`  A ) ) y  <->  <. x ,  y
>.  e.  ( R  i^i  (  _I  |`  A ) ) )
5 brin 3866 . . . . 5  |-  ( x ( R  i^i  (  _I  |`  A ) ) y  <->  ( x R y  /\  x (  _I  |`  A )
y ) )
64, 5bitr3i 184 . . . 4  |-  ( <.
x ,  y >.  e.  ( R  i^i  (  _I  |`  A ) )  <-> 
( x R y  /\  x (  _I  |`  A ) y ) )
7 vex 2618 . . . . . . . . 9  |-  y  e. 
_V
87brres 4685 . . . . . . . 8  |-  ( x (  _I  |`  A ) y  <->  ( x  _I  y  /\  x  e.  A ) )
9 poirr 4106 . . . . . . . . . . 11  |-  ( ( R  Po  A  /\  x  e.  A )  ->  -.  x R x )
107ideq 4554 . . . . . . . . . . . . 13  |-  ( x  _I  y  <->  x  =  y )
11 breq2 3823 . . . . . . . . . . . . 13  |-  ( x  =  y  ->  (
x R x  <->  x R
y ) )
1210, 11sylbi 119 . . . . . . . . . . . 12  |-  ( x  _I  y  ->  (
x R x  <->  x R
y ) )
1312notbid 625 . . . . . . . . . . 11  |-  ( x  _I  y  ->  ( -.  x R x  <->  -.  x R y ) )
149, 13syl5ibcom 153 . . . . . . . . . 10  |-  ( ( R  Po  A  /\  x  e.  A )  ->  ( x  _I  y  ->  -.  x R y ) )
1514expimpd 355 . . . . . . . . 9  |-  ( R  Po  A  ->  (
( x  e.  A  /\  x  _I  y
)  ->  -.  x R y ) )
1615ancomsd 265 . . . . . . . 8  |-  ( R  Po  A  ->  (
( x  _I  y  /\  x  e.  A
)  ->  -.  x R y ) )
178, 16syl5bi 150 . . . . . . 7  |-  ( R  Po  A  ->  (
x (  _I  |`  A ) y  ->  -.  x R y ) )
1817con2d 587 . . . . . 6  |-  ( R  Po  A  ->  (
x R y  ->  -.  x (  _I  |`  A ) y ) )
19 imnan 657 . . . . . 6  |-  ( ( x R y  ->  -.  x (  _I  |`  A ) y )  <->  -.  (
x R y  /\  x (  _I  |`  A ) y ) )
2018, 19sylib 120 . . . . 5  |-  ( R  Po  A  ->  -.  ( x R y  /\  x (  _I  |`  A ) y ) )
2120pm2.21d 582 . . . 4  |-  ( R  Po  A  ->  (
( x R y  /\  x (  _I  |`  A ) y )  ->  <. x ,  y
>.  e.  (/) ) )
226, 21syl5bi 150 . . 3  |-  ( R  Po  A  ->  ( <. x ,  y >.  e.  ( R  i^i  (  _I  |`  A ) )  ->  <. x ,  y
>.  e.  (/) ) )
233, 22relssdv 4496 . 2  |-  ( R  Po  A  ->  ( R  i^i  (  _I  |`  A ) )  C_  (/) )
24 ss0 3311 . 2  |-  ( ( R  i^i  (  _I  |`  A ) )  C_  (/) 
->  ( R  i^i  (  _I  |`  A ) )  =  (/) )
2523, 24syl 14 1  |-  ( R  Po  A  ->  ( R  i^i  (  _I  |`  A ) )  =  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1287    e. wcel 1436    i^i cin 2987    C_ wss 2988   (/)c0 3275   <.cop 3433   class class class wbr 3819    _I cid 4087    Po wpo 4093    |` cres 4411   Rel wrel 4414
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3930  ax-pow 3982  ax-pr 4008
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rex 2361  df-v 2617  df-dif 2990  df-un 2992  df-in 2994  df-ss 3001  df-nul 3276  df-pw 3416  df-sn 3436  df-pr 3437  df-op 3439  df-br 3820  df-opab 3874  df-id 4092  df-po 4095  df-xp 4415  df-rel 4416  df-res 4421
This theorem is referenced by: (None)
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