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Theorem fr2nr 4812
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 4642 . . . . . . 7  |-  { B ,  C }  e.  _V
21a1i 11 . . . . . 6  |-  ( ( R  Fr  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  { B ,  C }  e.  _V )
3 simpl 459 . . . . . 6  |-  ( ( R  Fr  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  R  Fr  A )
4 prssi 4128 . . . . . . 7  |-  ( ( B  e.  A  /\  C  e.  A )  ->  { B ,  C }  C_  A )
54adantl 468 . . . . . 6  |-  ( ( R  Fr  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  { B ,  C }  C_  A
)
6 prnzg 4092 . . . . . . 7  |-  ( B  e.  A  ->  { B ,  C }  =/=  (/) )
76ad2antrl 734 . . . . . 6  |-  ( ( R  Fr  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  { B ,  C }  =/=  (/) )
8 fri 4796 . . . . . 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 1269 . . . . 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 4406 . . . . . . . . 9  |-  ( y  =  B  ->  (
x R y  <->  x R B ) )
1110notbid 296 . . . . . . . 8  |-  ( y  =  B  ->  ( -.  x R y  <->  -.  x R B ) )
1211ralbidv 2827 . . . . . . 7  |-  ( y  =  B  ->  ( A. x  e.  { B ,  C }  -.  x R y  <->  A. x  e.  { B ,  C }  -.  x R B ) )
13 breq2 4406 . . . . . . . . 9  |-  ( y  =  C  ->  (
x R y  <->  x R C ) )
1413notbid 296 . . . . . . . 8  |-  ( y  =  C  ->  ( -.  x R y  <->  -.  x R C ) )
1514ralbidv 2827 . . . . . . 7  |-  ( y  =  C  ->  ( A. x  e.  { B ,  C }  -.  x R y  <->  A. x  e.  { B ,  C }  -.  x R C ) )
1612, 15rexprg 4022 . . . . . 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 468 . . . . 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 214 . . . 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 4079 . . . . . . 7  |-  ( C  e.  A  ->  C  e.  { B ,  C } )
2019ad2antll 735 . . . . . 6  |-  ( ( R  Fr  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  C  e.  { B ,  C } )
21 breq1 4405 . . . . . . . 8  |-  ( x  =  C  ->  (
x R B  <->  C R B ) )
2221notbid 296 . . . . . . 7  |-  ( x  =  C  ->  ( -.  x R B  <->  -.  C R B ) )
2322rspcv 3146 . . . . . 6  |-  ( C  e.  { B ,  C }  ->  ( A. x  e.  { B ,  C }  -.  x R B  ->  -.  C R B ) )
2420, 23syl 17 . . . . 5  |-  ( ( R  Fr  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  ( A. x  e.  { B ,  C }  -.  x R B  ->  -.  C R B ) )
25 prid1g 4078 . . . . . . 7  |-  ( B  e.  A  ->  B  e.  { B ,  C } )
2625ad2antrl 734 . . . . . 6  |-  ( ( R  Fr  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  B  e.  { B ,  C } )
27 breq1 4405 . . . . . . . 8  |-  ( x  =  B  ->  (
x R C  <->  B R C ) )
2827notbid 296 . . . . . . 7  |-  ( x  =  B  ->  ( -.  x R C  <->  -.  B R C ) )
2928rspcv 3146 . . . . . 6  |-  ( B  e.  { B ,  C }  ->  ( A. x  e.  { B ,  C }  -.  x R C  ->  -.  B R C ) )
3026, 29syl 17 . . . . 5  |-  ( ( R  Fr  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  ( A. x  e.  { B ,  C }  -.  x R C  ->  -.  B R C ) )
3124, 30orim12d 849 . . . 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 15 . . 3  |-  ( ( R  Fr  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  ( -.  C R B  \/  -.  B R C ) )
3332orcomd 390 . 2  |-  ( ( R  Fr  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  ( -.  B R C  \/  -.  C R B ) )
34 ianor 491 . 2  |-  ( -.  ( B R C  /\  C R B )  <->  ( -.  B R C  \/  -.  C R B ) )
3533, 34sylibr 216 1  |-  ( ( R  Fr  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  -.  ( B R C  /\  C R B ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 188    \/ wo 370    /\ wa 371    = wceq 1444    e. wcel 1887    =/= wne 2622   A.wral 2737   E.wrex 2738   _Vcvv 3045    C_ wss 3404   (/)c0 3731   {cpr 3970   class class class wbr 4402    Fr wfr 4790
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pr 4639
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-sbc 3268  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-sn 3969  df-pr 3971  df-op 3975  df-br 4403  df-fr 4793
This theorem is referenced by:  efrn2lp  4816  dfwe2  6608
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