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Theorem intasym 5388
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 5380 . . 3  |-  Rel  `' R
2 relin2 5127 . . 3  |-  ( Rel  `' R  ->  Rel  ( R  i^i  `' R ) )
3 ssrel 5097 . . 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 3692 . . . . 5  |-  ( <.
x ,  y >.  e.  ( R  i^i  `' R )  <->  ( <. x ,  y >.  e.  R  /\  <. x ,  y
>.  e.  `' R ) )
6 df-br 4454 . . . . . 6  |-  ( x R y  <->  <. x ,  y >.  e.  R
)
7 vex 3121 . . . . . . . 8  |-  x  e. 
_V
8 vex 3121 . . . . . . . 8  |-  y  e. 
_V
97, 8brcnv 5191 . . . . . . 7  |-  ( x `' R y  <->  y R x )
10 df-br 4454 . . . . . . 7  |-  ( x `' R y  <->  <. x ,  y >.  e.  `' R )
119, 10bitr3i 251 . . . . . 6  |-  ( y R x  <->  <. x ,  y >.  e.  `' R )
126, 11anbi12i 697 . . . . 5  |-  ( ( x R y  /\  y R x )  <->  ( <. x ,  y >.  e.  R  /\  <. x ,  y
>.  e.  `' R ) )
135, 12bitr4i 252 . . . 4  |-  ( <.
x ,  y >.  e.  ( R  i^i  `' R )  <->  ( x R y  /\  y R x ) )
14 df-br 4454 . . . . 5  |-  ( x  _I  y  <->  <. x ,  y >.  e.  _I  )
158ideq 5161 . . . . 5  |-  ( x  _I  y  <->  x  =  y )
1614, 15bitr3i 251 . . . 4  |-  ( <.
x ,  y >.  e.  _I  <->  x  =  y
)
1713, 16imbi12i 326 . . 3  |-  ( (
<. x ,  y >.  e.  ( R  i^i  `' R )  ->  <. x ,  y >.  e.  _I  ) 
<->  ( ( x R y  /\  y R x )  ->  x  =  y ) )
18172albii 1621 . 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 249 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 184    /\ wa 369   A.wal 1377    e. wcel 1767    i^i cin 3480    C_ wss 3481   <.cop 4039   class class class wbr 4453    _I cid 4796   `'ccnv 5004   Rel wrel 5010
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pr 4692
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-br 4454  df-opab 4512  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013
This theorem is referenced by: (None)
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