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Theorem intirr 5252
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 3533 . . . 4  |-  ( R  i^i  _I  )  =  (  _I  i^i  R
)
21eqeq1i 2443 . . 3  |-  ( ( R  i^i  _I  )  =  (/)  <->  (  _I  i^i  R )  =  (/) )
3 disj2 3675 . . 3  |-  ( (  _I  i^i  R )  =  (/)  <->  _I  C_  ( _V 
\  R ) )
4 reli 5002 . . . 4  |-  Rel  _I
5 ssrel 4964 . . . 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 263 . 2  |-  ( ( R  i^i  _I  )  =  (/)  <->  A. x A. y
( <. x ,  y
>.  e.  _I  ->  <. x ,  y >.  e.  ( _V  \  R ) ) )
8 equcom 1692 . . . . 5  |-  ( y  =  x  <->  x  =  y )
9 vex 2959 . . . . . 6  |-  y  e. 
_V
109ideq 5025 . . . . 5  |-  ( x  _I  y  <->  x  =  y )
11 df-br 4213 . . . . 5  |-  ( x  _I  y  <->  <. x ,  y >.  e.  _I  )
128, 10, 113bitr2i 265 . . . 4  |-  ( y  =  x  <->  <. x ,  y >.  e.  _I  )
13 opex 4427 . . . . . . 7  |-  <. x ,  y >.  e.  _V
1413biantrur 493 . . . . . 6  |-  ( -. 
<. x ,  y >.  e.  R  <->  ( <. x ,  y >.  e.  _V  /\ 
-.  <. x ,  y
>.  e.  R ) )
15 eldif 3330 . . . . . 6  |-  ( <.
x ,  y >.  e.  ( _V  \  R
)  <->  ( <. x ,  y >.  e.  _V  /\ 
-.  <. x ,  y
>.  e.  R ) )
1614, 15bitr4i 244 . . . . 5  |-  ( -. 
<. x ,  y >.  e.  R  <->  <. x ,  y
>.  e.  ( _V  \  R ) )
17 df-br 4213 . . . . 5  |-  ( x R y  <->  <. x ,  y >.  e.  R
)
1816, 17xchnxbir 301 . . . 4  |-  ( -.  x R y  <->  <. x ,  y >.  e.  ( _V  \  R ) )
1912, 18imbi12i 317 . . 3  |-  ( ( y  =  x  ->  -.  x R y )  <-> 
( <. x ,  y
>.  e.  _I  ->  <. x ,  y >.  e.  ( _V  \  R ) ) )
20192albii 1576 . 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 1629 . . . 4  |-  F/ y  -.  x R x
22 breq2 4216 . . . . 5  |-  ( y  =  x  ->  (
x R y  <->  x R x ) )
2322notbid 286 . . . 4  |-  ( y  =  x  ->  ( -.  x R y  <->  -.  x R x ) )
2421, 23equsal 1999 . . 3  |-  ( A. y ( y  =  x  ->  -.  x R y )  <->  -.  x R x )
2524albii 1575 . 2  |-  ( A. x A. y ( y  =  x  ->  -.  x R y )  <->  A. x  -.  x R x )
267, 20, 253bitr2i 265 1  |-  ( ( R  i^i  _I  )  =  (/)  <->  A. x  -.  x R x )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359   A.wal 1549    = wceq 1652    e. wcel 1725   _Vcvv 2956    \ cdif 3317    i^i cin 3319    C_ wss 3320   (/)c0 3628   <.cop 3817   class class class wbr 4212    _I cid 4493   Rel wrel 4883
This theorem is referenced by:  hartogslem1  7511  hausdiag  17677
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-br 4213  df-opab 4267  df-id 4498  df-xp 4884  df-rel 4885
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