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Theorem intasym 5060
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 5053 . . 3  |-  Rel  `' R
2 relin2 4806 . . 3  |-  ( Rel  `' R  ->  Rel  ( R  i^i  `' R ) )
3 ssrel 4778 . . 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 9 . 2  |-  ( ( R  i^i  `' R
)  C_  _I  <->  A. x A. y ( <. x ,  y >.  e.  ( R  i^i  `' R
)  ->  <. x ,  y >.  e.  _I  ) )
5 elin 3360 . . . . 5  |-  ( <.
x ,  y >.  e.  ( R  i^i  `' R )  <->  ( <. x ,  y >.  e.  R  /\  <. x ,  y
>.  e.  `' R ) )
6 df-br 4026 . . . . . 6  |-  ( x R y  <->  <. x ,  y >.  e.  R
)
7 vex 2793 . . . . . . . 8  |-  x  e. 
_V
8 vex 2793 . . . . . . . 8  |-  y  e. 
_V
97, 8brcnv 4866 . . . . . . 7  |-  ( x `' R y  <->  y R x )
10 df-br 4026 . . . . . . 7  |-  ( x `' R y  <->  <. x ,  y >.  e.  `' R )
119, 10bitr3i 242 . . . . . 6  |-  ( y R x  <->  <. x ,  y >.  e.  `' R )
126, 11anbi12i 678 . . . . 5  |-  ( ( x R y  /\  y R x )  <->  ( <. x ,  y >.  e.  R  /\  <. x ,  y
>.  e.  `' R ) )
135, 12bitr4i 243 . . . 4  |-  ( <.
x ,  y >.  e.  ( R  i^i  `' R )  <->  ( x R y  /\  y R x ) )
14 df-br 4026 . . . . 5  |-  ( x  _I  y  <->  <. x ,  y >.  e.  _I  )
158ideq 4838 . . . . 5  |-  ( x  _I  y  <->  x  =  y )
1614, 15bitr3i 242 . . . 4  |-  ( <.
x ,  y >.  e.  _I  <->  x  =  y
)
1713, 16imbi12i 316 . . 3  |-  ( (
<. x ,  y >.  e.  ( R  i^i  `' R )  ->  <. x ,  y >.  e.  _I  ) 
<->  ( ( x R y  /\  y R x )  ->  x  =  y ) )
18172albii 1556 . 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 240 1  |-  ( ( R  i^i  `' R
)  C_  _I  <->  A. x A. y ( ( x R y  /\  y R x )  ->  x  =  y )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   A.wal 1529    = wceq 1625    e. wcel 1686    i^i cin 3153    C_ wss 3154   <.cop 3645   class class class wbr 4025    _I cid 4306   `'ccnv 4690   Rel wrel 4696
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  df-cnv 4699
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