MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  frsn Structured version   Unicode version

Theorem frsn 4948
Description: Founded relation on a singleton. (Contributed by Mario Carneiro, 28-Dec-2014.) (Revised by Mario Carneiro, 23-Apr-2015.)
Assertion
Ref Expression
frsn  |-  ( Rel 
R  ->  ( R  Fr  { A }  <->  -.  A R A ) )

Proof of Theorem frsn
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-fr 4541 . . . 4  |-  ( R  Fr  { A }  <->  A. x ( ( x 
C_  { A }  /\  x  =/=  (/) )  ->  E. y  e.  x  A. z  e.  x  -.  z R y ) )
2 df-ne 2601 . . . . . . . . . 10  |-  ( x  =/=  (/)  <->  -.  x  =  (/) )
3 simpr 448 . . . . . . . . . . . 12  |-  ( ( ( Rel  R  /\  A  e.  _V )  /\  x  C_  { A } )  ->  x  C_ 
{ A } )
4 sssn 3957 . . . . . . . . . . . 12  |-  ( x 
C_  { A }  <->  ( x  =  (/)  \/  x  =  { A } ) )
53, 4sylib 189 . . . . . . . . . . 11  |-  ( ( ( Rel  R  /\  A  e.  _V )  /\  x  C_  { A } )  ->  (
x  =  (/)  \/  x  =  { A } ) )
65ord 367 . . . . . . . . . 10  |-  ( ( ( Rel  R  /\  A  e.  _V )  /\  x  C_  { A } )  ->  ( -.  x  =  (/)  ->  x  =  { A } ) )
72, 6syl5bi 209 . . . . . . . . 9  |-  ( ( ( Rel  R  /\  A  e.  _V )  /\  x  C_  { A } )  ->  (
x  =/=  (/)  ->  x  =  { A } ) )
87impr 603 . . . . . . . 8  |-  ( ( ( Rel  R  /\  A  e.  _V )  /\  ( x  C_  { A }  /\  x  =/=  (/) ) )  ->  x  =  { A } )
9 eqimss 3400 . . . . . . . . . 10  |-  ( x  =  { A }  ->  x  C_  { A } )
109adantl 453 . . . . . . . . 9  |-  ( ( ( Rel  R  /\  A  e.  _V )  /\  x  =  { A } )  ->  x  C_ 
{ A } )
11 simpr 448 . . . . . . . . . 10  |-  ( ( ( Rel  R  /\  A  e.  _V )  /\  x  =  { A } )  ->  x  =  { A } )
12 snnzg 3921 . . . . . . . . . . 11  |-  ( A  e.  _V  ->  { A }  =/=  (/) )
1312ad2antlr 708 . . . . . . . . . 10  |-  ( ( ( Rel  R  /\  A  e.  _V )  /\  x  =  { A } )  ->  { A }  =/=  (/) )
1411, 13eqnetrd 2619 . . . . . . . . 9  |-  ( ( ( Rel  R  /\  A  e.  _V )  /\  x  =  { A } )  ->  x  =/=  (/) )
1510, 14jca 519 . . . . . . . 8  |-  ( ( ( Rel  R  /\  A  e.  _V )  /\  x  =  { A } )  ->  (
x  C_  { A }  /\  x  =/=  (/) ) )
168, 15impbida 806 . . . . . . 7  |-  ( ( Rel  R  /\  A  e.  _V )  ->  (
( x  C_  { A }  /\  x  =/=  (/) )  <->  x  =  { A } ) )
1716imbi1d 309 . . . . . 6  |-  ( ( Rel  R  /\  A  e.  _V )  ->  (
( ( x  C_  { A }  /\  x  =/=  (/) )  ->  E. y  e.  x  A. z  e.  x  -.  z R y )  <->  ( x  =  { A }  ->  E. y  e.  x  A. z  e.  x  -.  z R y ) ) )
1817albidv 1635 . . . . 5  |-  ( ( Rel  R  /\  A  e.  _V )  ->  ( A. x ( ( x 
C_  { A }  /\  x  =/=  (/) )  ->  E. y  e.  x  A. z  e.  x  -.  z R y )  <->  A. x ( x  =  { A }  ->  E. y  e.  x  A. z  e.  x  -.  z R y ) ) )
19 snex 4405 . . . . . 6  |-  { A }  e.  _V
20 raleq 2904 . . . . . . 7  |-  ( x  =  { A }  ->  ( A. z  e.  x  -.  z R y  <->  A. z  e.  { A }  -.  z R y ) )
2120rexeqbi1dv 2913 . . . . . 6  |-  ( x  =  { A }  ->  ( E. y  e.  x  A. z  e.  x  -.  z R y  <->  E. y  e.  { A } A. z  e. 
{ A }  -.  z R y ) )
2219, 21ceqsalv 2982 . . . . 5  |-  ( A. x ( x  =  { A }  ->  E. y  e.  x  A. z  e.  x  -.  z R y )  <->  E. y  e.  { A } A. z  e.  { A }  -.  z R y )
2318, 22syl6bb 253 . . . 4  |-  ( ( Rel  R  /\  A  e.  _V )  ->  ( A. x ( ( x 
C_  { A }  /\  x  =/=  (/) )  ->  E. y  e.  x  A. z  e.  x  -.  z R y )  <->  E. y  e.  { A } A. z  e.  { A }  -.  z R y ) )
241, 23syl5bb 249 . . 3  |-  ( ( Rel  R  /\  A  e.  _V )  ->  ( R  Fr  { A } 
<->  E. y  e.  { A } A. z  e. 
{ A }  -.  z R y ) )
25 breq2 4216 . . . . . . . 8  |-  ( y  =  A  ->  (
z R y  <->  z R A ) )
2625notbid 286 . . . . . . 7  |-  ( y  =  A  ->  ( -.  z R y  <->  -.  z R A ) )
2726ralbidv 2725 . . . . . 6  |-  ( y  =  A  ->  ( A. z  e.  { A }  -.  z R y  <->  A. z  e.  { A }  -.  z R A ) )
2827rexsng 3847 . . . . 5  |-  ( A  e.  _V  ->  ( E. y  e.  { A } A. z  e.  { A }  -.  z R y  <->  A. z  e.  { A }  -.  z R A ) )
29 breq1 4215 . . . . . . 7  |-  ( z  =  A  ->  (
z R A  <->  A R A ) )
3029notbid 286 . . . . . 6  |-  ( z  =  A  ->  ( -.  z R A  <->  -.  A R A ) )
3130ralsng 3846 . . . . 5  |-  ( A  e.  _V  ->  ( A. z  e.  { A }  -.  z R A  <->  -.  A R A ) )
3228, 31bitrd 245 . . . 4  |-  ( A  e.  _V  ->  ( E. y  e.  { A } A. z  e.  { A }  -.  z R y  <->  -.  A R A ) )
3332adantl 453 . . 3  |-  ( ( Rel  R  /\  A  e.  _V )  ->  ( E. y  e.  { A } A. z  e.  { A }  -.  z R y  <->  -.  A R A ) )
3424, 33bitrd 245 . 2  |-  ( ( Rel  R  /\  A  e.  _V )  ->  ( R  Fr  { A } 
<->  -.  A R A ) )
35 snprc 3871 . . . . 5  |-  ( -.  A  e.  _V  <->  { A }  =  (/) )
36 fr0 4561 . . . . . 6  |-  R  Fr  (/)
37 freq2 4553 . . . . . 6  |-  ( { A }  =  (/)  ->  ( R  Fr  { A }  <->  R  Fr  (/) ) )
3836, 37mpbiri 225 . . . . 5  |-  ( { A }  =  (/)  ->  R  Fr  { A } )
3935, 38sylbi 188 . . . 4  |-  ( -.  A  e.  _V  ->  R  Fr  { A }
)
4039adantl 453 . . 3  |-  ( ( Rel  R  /\  -.  A  e.  _V )  ->  R  Fr  { A } )
41 brrelex 4916 . . . . 5  |-  ( ( Rel  R  /\  A R A )  ->  A  e.  _V )
4241ex 424 . . . 4  |-  ( Rel 
R  ->  ( A R A  ->  A  e. 
_V ) )
4342con3and 429 . . 3  |-  ( ( Rel  R  /\  -.  A  e.  _V )  ->  -.  A R A )
4440, 432thd 232 . 2  |-  ( ( Rel  R  /\  -.  A  e.  _V )  ->  ( R  Fr  { A }  <->  -.  A R A ) )
4534, 44pm2.61dan 767 1  |-  ( Rel 
R  ->  ( R  Fr  { A }  <->  -.  A R A ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359   A.wal 1549    = wceq 1652    e. wcel 1725    =/= wne 2599   A.wral 2705   E.wrex 2706   _Vcvv 2956    C_ wss 3320   (/)c0 3628   {csn 3814   class class class wbr 4212    Fr wfr 4538   Rel wrel 4883
This theorem is referenced by:  wesn  4949
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-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-sbc 3162  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-fr 4541  df-xp 4884  df-rel 4885
  Copyright terms: Public domain W3C validator