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Theorem intirr 5063
Description: Two ways of saying a relation is irreflexive. Definition of irreflexivity in [Schechter] p. 51. (Contributed by NM, 9-Sep-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
Assertion
Ref Expression
intirr  |-  ( ( R  i^i  _I  )  =  (/)  <->  A. x  -.  x R x )
Distinct variable group:    x, R

Proof of Theorem intirr
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 incom 3363 . . . 4  |-  ( R  i^i  _I  )  =  (  _I  i^i  R
)
21eqeq1i 2292 . . 3  |-  ( ( R  i^i  _I  )  =  (/)  <->  (  _I  i^i  R )  =  (/) )
3 disj2 3504 . . 3  |-  ( (  _I  i^i  R )  =  (/)  <->  _I  C_  ( _V 
\  R ) )
4 reli 4815 . . . 4  |-  Rel  _I
5 ssrel 4778 . . . 4  |-  ( Rel 
_I  ->  (  _I  C_  ( _V  \  R )  <->  A. x A. y (
<. x ,  y >.  e.  _I  ->  <. x ,  y >.  e.  ( _V  \  R ) ) ) )
64, 5ax-mp 8 . . 3  |-  (  _I  C_  ( _V  \  R
)  <->  A. x A. y
( <. x ,  y
>.  e.  _I  ->  <. x ,  y >.  e.  ( _V  \  R ) ) )
72, 3, 63bitri 262 . 2  |-  ( ( R  i^i  _I  )  =  (/)  <->  A. x A. y
( <. x ,  y
>.  e.  _I  ->  <. x ,  y >.  e.  ( _V  \  R ) ) )
8 eqcom 2287 . . . . 5  |-  ( y  =  x  <->  x  =  y )
9 vex 2793 . . . . . 6  |-  y  e. 
_V
109ideq 4838 . . . . 5  |-  ( x  _I  y  <->  x  =  y )
11 df-br 4026 . . . . 5  |-  ( x  _I  y  <->  <. x ,  y >.  e.  _I  )
128, 10, 113bitr2i 264 . . . 4  |-  ( y  =  x  <->  <. x ,  y >.  e.  _I  )
13 opex 4239 . . . . . . 7  |-  <. x ,  y >.  e.  _V
1413biantrur 492 . . . . . 6  |-  ( -. 
<. x ,  y >.  e.  R  <->  ( <. x ,  y >.  e.  _V  /\ 
-.  <. x ,  y
>.  e.  R ) )
15 eldif 3164 . . . . . 6  |-  ( <.
x ,  y >.  e.  ( _V  \  R
)  <->  ( <. x ,  y >.  e.  _V  /\ 
-.  <. x ,  y
>.  e.  R ) )
1614, 15bitr4i 243 . . . . 5  |-  ( -. 
<. x ,  y >.  e.  R  <->  <. x ,  y
>.  e.  ( _V  \  R ) )
17 df-br 4026 . . . . 5  |-  ( x R y  <->  <. x ,  y >.  e.  R
)
1816, 17xchnxbir 300 . . . 4  |-  ( -.  x R y  <->  <. x ,  y >.  e.  ( _V  \  R ) )
1912, 18imbi12i 316 . . 3  |-  ( ( y  =  x  ->  -.  x R y )  <-> 
( <. x ,  y
>.  e.  _I  ->  <. x ,  y >.  e.  ( _V  \  R ) ) )
20192albii 1556 . 2  |-  ( A. x A. y ( y  =  x  ->  -.  x R y )  <->  A. x A. y ( <. x ,  y >.  e.  _I  -> 
<. x ,  y >.  e.  ( _V  \  R
) ) )
21 nfv 1607 . . . 4  |-  F/ y  -.  x R x
22 breq2 4029 . . . . 5  |-  ( y  =  x  ->  (
x R y  <->  x R x ) )
2322notbid 285 . . . 4  |-  ( y  =  x  ->  ( -.  x R y  <->  -.  x R x ) )
2421, 23equsal 1902 . . 3  |-  ( A. y ( y  =  x  ->  -.  x R y )  <->  -.  x R x )
2524albii 1555 . 2  |-  ( A. x A. y ( y  =  x  ->  -.  x R y )  <->  A. x  -.  x R x )
267, 20, 253bitr2i 264 1  |-  ( ( R  i^i  _I  )  =  (/)  <->  A. x  -.  x R x )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358   A.wal 1529    = wceq 1625    e. wcel 1686   _Vcvv 2790    \ cdif 3151    i^i cin 3153    C_ wss 3154   (/)c0 3457   <.cop 3645   class class class wbr 4025    _I cid 4306   Rel wrel 4696
This theorem is referenced by:  hartogslem1  7259  hausdiag  17341
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pr 4216
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-rab 2554  df-v 2792  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-sn 3648  df-pr 3649  df-op 3651  df-br 4026  df-opab 4080  df-id 4311  df-xp 4697  df-rel 4698
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