MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fr2nr Unicode version

Theorem fr2nr 4371
Description: A well-founded relation has no 2-cycle loops. Special case of Proposition 6.23 of [TakeutiZaring] p. 30. (Contributed by NM, 30-May-1994.) (Revised by Mario Carneiro, 22-Jun-2015.)
Assertion
Ref Expression
fr2nr  |-  ( ( R  Fr  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  -.  ( B R C  /\  C R B ) )

Proof of Theorem fr2nr
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 prex 4217 . . . . . . 7  |-  { B ,  C }  e.  _V
21a1i 10 . . . . . 6  |-  ( ( R  Fr  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  { B ,  C }  e.  _V )
3 simpl 443 . . . . . 6  |-  ( ( R  Fr  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  R  Fr  A )
4 prssi 3771 . . . . . . 7  |-  ( ( B  e.  A  /\  C  e.  A )  ->  { B ,  C }  C_  A )
54adantl 452 . . . . . 6  |-  ( ( R  Fr  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  { B ,  C }  C_  A
)
6 prnzg 3746 . . . . . . 7  |-  ( B  e.  A  ->  { B ,  C }  =/=  (/) )
76ad2antrl 708 . . . . . 6  |-  ( ( R  Fr  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  { B ,  C }  =/=  (/) )
8 fri 4355 . . . . . 6  |-  ( ( ( { B ,  C }  e.  _V  /\  R  Fr  A )  /\  ( { B ,  C }  C_  A  /\  { B ,  C }  =/=  (/) ) )  ->  E. y  e.  { B ,  C } A. x  e.  { B ,  C }  -.  x R y )
92, 3, 5, 7, 8syl22anc 1183 . . . . 5  |-  ( ( R  Fr  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  E. y  e.  { B ,  C } A. x  e.  { B ,  C }  -.  x R y )
10 breq2 4027 . . . . . . . . 9  |-  ( y  =  B  ->  (
x R y  <->  x R B ) )
1110notbid 285 . . . . . . . 8  |-  ( y  =  B  ->  ( -.  x R y  <->  -.  x R B ) )
1211ralbidv 2563 . . . . . . 7  |-  ( y  =  B  ->  ( A. x  e.  { B ,  C }  -.  x R y  <->  A. x  e.  { B ,  C }  -.  x R B ) )
13 breq2 4027 . . . . . . . . 9  |-  ( y  =  C  ->  (
x R y  <->  x R C ) )
1413notbid 285 . . . . . . . 8  |-  ( y  =  C  ->  ( -.  x R y  <->  -.  x R C ) )
1514ralbidv 2563 . . . . . . 7  |-  ( y  =  C  ->  ( A. x  e.  { B ,  C }  -.  x R y  <->  A. x  e.  { B ,  C }  -.  x R C ) )
1612, 15rexprg 3683 . . . . . 6  |-  ( ( B  e.  A  /\  C  e.  A )  ->  ( E. y  e. 
{ B ,  C } A. x  e.  { B ,  C }  -.  x R y  <->  ( A. x  e.  { B ,  C }  -.  x R B  \/  A. x  e.  { B ,  C }  -.  x R C ) ) )
1716adantl 452 . . . . 5  |-  ( ( R  Fr  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  ( E. y  e.  { B ,  C } A. x  e.  { B ,  C }  -.  x R y  <-> 
( A. x  e. 
{ B ,  C }  -.  x R B  \/  A. x  e. 
{ B ,  C }  -.  x R C ) ) )
189, 17mpbid 201 . . . 4  |-  ( ( R  Fr  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  ( A. x  e.  { B ,  C }  -.  x R B  \/  A. x  e.  { B ,  C }  -.  x R C ) )
19 prid2g 3733 . . . . . . 7  |-  ( C  e.  A  ->  C  e.  { B ,  C } )
2019ad2antll 709 . . . . . 6  |-  ( ( R  Fr  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  C  e.  { B ,  C } )
21 breq1 4026 . . . . . . . 8  |-  ( x  =  C  ->  (
x R B  <->  C R B ) )
2221notbid 285 . . . . . . 7  |-  ( x  =  C  ->  ( -.  x R B  <->  -.  C R B ) )
2322rspcv 2880 . . . . . 6  |-  ( C  e.  { B ,  C }  ->  ( A. x  e.  { B ,  C }  -.  x R B  ->  -.  C R B ) )
2420, 23syl 15 . . . . 5  |-  ( ( R  Fr  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  ( A. x  e.  { B ,  C }  -.  x R B  ->  -.  C R B ) )
25 prid1g 3732 . . . . . . 7  |-  ( B  e.  A  ->  B  e.  { B ,  C } )
2625ad2antrl 708 . . . . . 6  |-  ( ( R  Fr  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  B  e.  { B ,  C } )
27 breq1 4026 . . . . . . . 8  |-  ( x  =  B  ->  (
x R C  <->  B R C ) )
2827notbid 285 . . . . . . 7  |-  ( x  =  B  ->  ( -.  x R C  <->  -.  B R C ) )
2928rspcv 2880 . . . . . 6  |-  ( B  e.  { B ,  C }  ->  ( A. x  e.  { B ,  C }  -.  x R C  ->  -.  B R C ) )
3026, 29syl 15 . . . . 5  |-  ( ( R  Fr  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  ( A. x  e.  { B ,  C }  -.  x R C  ->  -.  B R C ) )
3124, 30orim12d 811 . . . 4  |-  ( ( R  Fr  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  (
( A. x  e. 
{ B ,  C }  -.  x R B  \/  A. x  e. 
{ B ,  C }  -.  x R C )  ->  ( -.  C R B  \/  -.  B R C ) ) )
3218, 31mpd 14 . . 3  |-  ( ( R  Fr  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  ( -.  C R B  \/  -.  B R C ) )
3332orcomd 377 . 2  |-  ( ( R  Fr  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  ( -.  B R C  \/  -.  C R B ) )
34 ianor 474 . 2  |-  ( -.  ( B R C  /\  C R B )  <->  ( -.  B R C  \/  -.  C R B ) )
3533, 34sylibr 203 1  |-  ( ( R  Fr  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  -.  ( B R C  /\  C R B ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   E.wrex 2544   _Vcvv 2788    C_ wss 3152   (/)c0 3455   {cpr 3641   class class class wbr 4023    Fr wfr 4349
This theorem is referenced by:  efrn2lp  4375  dfwe2  4573
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-fr 4352
  Copyright terms: Public domain W3C validator