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Theorem poirr2 5221
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 5137 . . . 4  |-  Rel  (  _I  |`  A )
2 relin2 4956 . . . 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 4177 . . . . 5  |-  ( x ( R  i^i  (  _I  |`  A ) ) y  <->  <. x ,  y
>.  e.  ( R  i^i  (  _I  |`  A ) ) )
5 brin 4223 . . . . 5  |-  ( x ( R  i^i  (  _I  |`  A ) ) y  <->  ( x R y  /\  x (  _I  |`  A )
y ) )
64, 5bitr3i 243 . . . 4  |-  ( <.
x ,  y >.  e.  ( R  i^i  (  _I  |`  A ) )  <-> 
( x R y  /\  x (  _I  |`  A ) y ) )
7 vex 2923 . . . . . . . . 9  |-  y  e. 
_V
87brres 5115 . . . . . . . 8  |-  ( x (  _I  |`  A ) y  <->  ( x  _I  y  /\  x  e.  A ) )
9 poirr 4478 . . . . . . . . . . 11  |-  ( ( R  Po  A  /\  x  e.  A )  ->  -.  x R x )
107ideq 4988 . . . . . . . . . . . . 13  |-  ( x  _I  y  <->  x  =  y )
11 breq2 4180 . . . . . . . . . . . . 13  |-  ( x  =  y  ->  (
x R x  <->  x R
y ) )
1210, 11sylbi 188 . . . . . . . . . . . 12  |-  ( x  _I  y  ->  (
x R x  <->  x R
y ) )
1312notbid 286 . . . . . . . . . . 11  |-  ( x  _I  y  ->  ( -.  x R x  <->  -.  x R y ) )
149, 13syl5ibcom 212 . . . . . . . . . 10  |-  ( ( R  Po  A  /\  x  e.  A )  ->  ( x  _I  y  ->  -.  x R y ) )
1514expimpd 587 . . . . . . . . 9  |-  ( R  Po  A  ->  (
( x  e.  A  /\  x  _I  y
)  ->  -.  x R y ) )
1615ancomsd 441 . . . . . . . 8  |-  ( R  Po  A  ->  (
( x  _I  y  /\  x  e.  A
)  ->  -.  x R y ) )
178, 16syl5bi 209 . . . . . . 7  |-  ( R  Po  A  ->  (
x (  _I  |`  A ) y  ->  -.  x R y ) )
1817con2d 109 . . . . . 6  |-  ( R  Po  A  ->  (
x R y  ->  -.  x (  _I  |`  A ) y ) )
19 imnan 412 . . . . . 6  |-  ( ( x R y  ->  -.  x (  _I  |`  A ) y )  <->  -.  (
x R y  /\  x (  _I  |`  A ) y ) )
2018, 19sylib 189 . . . . 5  |-  ( R  Po  A  ->  -.  ( x R y  /\  x (  _I  |`  A ) y ) )
2120pm2.21d 100 . . . 4  |-  ( R  Po  A  ->  (
( x R y  /\  x (  _I  |`  A ) y )  ->  <. x ,  y
>.  e.  (/) ) )
226, 21syl5bi 209 . . 3  |-  ( R  Po  A  ->  ( <. x ,  y >.  e.  ( R  i^i  (  _I  |`  A ) )  ->  <. x ,  y
>.  e.  (/) ) )
233, 22relssdv 4931 . 2  |-  ( R  Po  A  ->  ( R  i^i  (  _I  |`  A ) )  C_  (/) )
24 ss0 3622 . 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 set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721    i^i cin 3283    C_ wss 3284   (/)c0 3592   <.cop 3781   class class class wbr 4176    _I cid 4457    Po wpo 4465    |` cres 4843   Rel wrel 4846
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-sep 4294  ax-nul 4302  ax-pr 4367
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-rab 2679  df-v 2922  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-sn 3784  df-pr 3785  df-op 3787  df-br 4177  df-opab 4231  df-id 4462  df-po 4467  df-xp 4847  df-rel 4848  df-res 4853
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