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Theorem intasym 5235
Description: Two ways of saying a relation is antisymmetric. Definition of antisymmetry in [Schechter] p. 51. (Contributed by NM, 9-Sep-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
Assertion
Ref Expression
intasym  |-  ( ( R  i^i  `' R
)  C_  _I  <->  A. x A. y ( ( x R y  /\  y R x )  ->  x  =  y )
)
Distinct variable group:    x, y, R

Proof of Theorem intasym
StepHypRef Expression
1 relcnv 5227 . . 3  |-  Rel  `' R
2 relin2 4972 . . 3  |-  ( Rel  `' R  ->  Rel  ( R  i^i  `' R ) )
3 ssrel 4943 . . 3  |-  ( Rel  ( R  i^i  `' R )  ->  (
( R  i^i  `' R )  C_  _I  <->  A. x A. y (
<. x ,  y >.  e.  ( R  i^i  `' R )  ->  <. x ,  y >.  e.  _I  ) ) )
41, 2, 3mp2b 10 . 2  |-  ( ( R  i^i  `' R
)  C_  _I  <->  A. x A. y ( <. x ,  y >.  e.  ( R  i^i  `' R
)  ->  <. x ,  y >.  e.  _I  ) )
5 elin 3655 . . . . 5  |-  ( <.
x ,  y >.  e.  ( R  i^i  `' R )  <->  ( <. x ,  y >.  e.  R  /\  <. x ,  y
>.  e.  `' R ) )
6 df-br 4427 . . . . . 6  |-  ( x R y  <->  <. x ,  y >.  e.  R
)
7 vex 3090 . . . . . . . 8  |-  x  e. 
_V
8 vex 3090 . . . . . . . 8  |-  y  e. 
_V
97, 8brcnv 5037 . . . . . . 7  |-  ( x `' R y  <->  y R x )
10 df-br 4427 . . . . . . 7  |-  ( x `' R y  <->  <. x ,  y >.  e.  `' R )
119, 10bitr3i 254 . . . . . 6  |-  ( y R x  <->  <. x ,  y >.  e.  `' R )
126, 11anbi12i 701 . . . . 5  |-  ( ( x R y  /\  y R x )  <->  ( <. x ,  y >.  e.  R  /\  <. x ,  y
>.  e.  `' R ) )
135, 12bitr4i 255 . . . 4  |-  ( <.
x ,  y >.  e.  ( R  i^i  `' R )  <->  ( x R y  /\  y R x ) )
14 df-br 4427 . . . . 5  |-  ( x  _I  y  <->  <. x ,  y >.  e.  _I  )
158ideq 5007 . . . . 5  |-  ( x  _I  y  <->  x  =  y )
1614, 15bitr3i 254 . . . 4  |-  ( <.
x ,  y >.  e.  _I  <->  x  =  y
)
1713, 16imbi12i 327 . . 3  |-  ( (
<. x ,  y >.  e.  ( R  i^i  `' R )  ->  <. x ,  y >.  e.  _I  ) 
<->  ( ( x R y  /\  y R x )  ->  x  =  y ) )
18172albii 1688 . 2  |-  ( A. x A. y ( <.
x ,  y >.  e.  ( R  i^i  `' R )  ->  <. x ,  y >.  e.  _I  ) 
<-> 
A. x A. y
( ( x R y  /\  y R x )  ->  x  =  y ) )
194, 18bitri 252 1  |-  ( ( R  i^i  `' R
)  C_  _I  <->  A. x A. y ( ( x R y  /\  y R x )  ->  x  =  y )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370   A.wal 1435    e. wcel 1870    i^i cin 3441    C_ wss 3442   <.cop 4008   class class class wbr 4426    _I cid 4764   `'ccnv 4853   Rel wrel 4859
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pr 4661
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-br 4427  df-opab 4485  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862
This theorem is referenced by: (None)
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