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Theorem poirr2 4931
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 4847 . . . 4  |-  Rel  (  _I  |`  A )
2 relin2 4658 . . . 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 3930 . . . . 5  |-  ( x ( R  i^i  (  _I  |`  A ) ) y  <->  <. x ,  y
>.  e.  ( R  i^i  (  _I  |`  A ) ) )
5 brin 3980 . . . . 5  |-  ( x ( R  i^i  (  _I  |`  A ) ) y  <->  ( x R y  /\  x (  _I  |`  A )
y ) )
64, 5bitr3i 185 . . . 4  |-  ( <.
x ,  y >.  e.  ( R  i^i  (  _I  |`  A ) )  <-> 
( x R y  /\  x (  _I  |`  A ) y ) )
7 vex 2689 . . . . . . . . 9  |-  y  e. 
_V
87brres 4825 . . . . . . . 8  |-  ( x (  _I  |`  A ) y  <->  ( x  _I  y  /\  x  e.  A ) )
9 poirr 4229 . . . . . . . . . . 11  |-  ( ( R  Po  A  /\  x  e.  A )  ->  -.  x R x )
107ideq 4691 . . . . . . . . . . . . 13  |-  ( x  _I  y  <->  x  =  y )
11 breq2 3933 . . . . . . . . . . . . 13  |-  ( x  =  y  ->  (
x R x  <->  x R
y ) )
1210, 11sylbi 120 . . . . . . . . . . . 12  |-  ( x  _I  y  ->  (
x R x  <->  x R
y ) )
1312notbid 656 . . . . . . . . . . 11  |-  ( x  _I  y  ->  ( -.  x R x  <->  -.  x R y ) )
149, 13syl5ibcom 154 . . . . . . . . . 10  |-  ( ( R  Po  A  /\  x  e.  A )  ->  ( x  _I  y  ->  -.  x R y ) )
1514expimpd 360 . . . . . . . . 9  |-  ( R  Po  A  ->  (
( x  e.  A  /\  x  _I  y
)  ->  -.  x R y ) )
1615ancomsd 267 . . . . . . . 8  |-  ( R  Po  A  ->  (
( x  _I  y  /\  x  e.  A
)  ->  -.  x R y ) )
178, 16syl5bi 151 . . . . . . 7  |-  ( R  Po  A  ->  (
x (  _I  |`  A ) y  ->  -.  x R y ) )
1817con2d 613 . . . . . 6  |-  ( R  Po  A  ->  (
x R y  ->  -.  x (  _I  |`  A ) y ) )
19 imnan 679 . . . . . 6  |-  ( ( x R y  ->  -.  x (  _I  |`  A ) y )  <->  -.  (
x R y  /\  x (  _I  |`  A ) y ) )
2018, 19sylib 121 . . . . 5  |-  ( R  Po  A  ->  -.  ( x R y  /\  x (  _I  |`  A ) y ) )
2120pm2.21d 608 . . . 4  |-  ( R  Po  A  ->  (
( x R y  /\  x (  _I  |`  A ) y )  ->  <. x ,  y
>.  e.  (/) ) )
226, 21syl5bi 151 . . 3  |-  ( R  Po  A  ->  ( <. x ,  y >.  e.  ( R  i^i  (  _I  |`  A ) )  ->  <. x ,  y
>.  e.  (/) ) )
233, 22relssdv 4631 . 2  |-  ( R  Po  A  ->  ( R  i^i  (  _I  |`  A ) )  C_  (/) )
24 ss0 3403 . 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 103    <-> wb 104    = wceq 1331    e. wcel 1480    i^i cin 3070    C_ wss 3071   (/)c0 3363   <.cop 3530   class class class wbr 3929    _I cid 4210    Po wpo 4216    |` cres 4541   Rel wrel 4544
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930  df-opab 3990  df-id 4215  df-po 4218  df-xp 4545  df-rel 4546  df-res 4551
This theorem is referenced by: (None)
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