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Theorem fr2nr 4800
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 4632 . . . . . . 7  |-  { B ,  C }  e.  _V
21a1i 11 . . . . . 6  |-  ( ( R  Fr  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  { B ,  C }  e.  _V )
3 simpl 455 . . . . . 6  |-  ( ( R  Fr  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  R  Fr  A )
4 prssi 4127 . . . . . . 7  |-  ( ( B  e.  A  /\  C  e.  A )  ->  { B ,  C }  C_  A )
54adantl 464 . . . . . 6  |-  ( ( R  Fr  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  { B ,  C }  C_  A
)
6 prnzg 4091 . . . . . . 7  |-  ( B  e.  A  ->  { B ,  C }  =/=  (/) )
76ad2antrl 726 . . . . . 6  |-  ( ( R  Fr  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  { B ,  C }  =/=  (/) )
8 fri 4784 . . . . . 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 1231 . . . . 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 4398 . . . . . . . . 9  |-  ( y  =  B  ->  (
x R y  <->  x R B ) )
1110notbid 292 . . . . . . . 8  |-  ( y  =  B  ->  ( -.  x R y  <->  -.  x R B ) )
1211ralbidv 2842 . . . . . . 7  |-  ( y  =  B  ->  ( A. x  e.  { B ,  C }  -.  x R y  <->  A. x  e.  { B ,  C }  -.  x R B ) )
13 breq2 4398 . . . . . . . . 9  |-  ( y  =  C  ->  (
x R y  <->  x R C ) )
1413notbid 292 . . . . . . . 8  |-  ( y  =  C  ->  ( -.  x R y  <->  -.  x R C ) )
1514ralbidv 2842 . . . . . . 7  |-  ( y  =  C  ->  ( A. x  e.  { B ,  C }  -.  x R y  <->  A. x  e.  { B ,  C }  -.  x R C ) )
1612, 15rexprg 4021 . . . . . 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 464 . . . . 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 210 . . . 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 4078 . . . . . . 7  |-  ( C  e.  A  ->  C  e.  { B ,  C } )
2019ad2antll 727 . . . . . 6  |-  ( ( R  Fr  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  C  e.  { B ,  C } )
21 breq1 4397 . . . . . . . 8  |-  ( x  =  C  ->  (
x R B  <->  C R B ) )
2221notbid 292 . . . . . . 7  |-  ( x  =  C  ->  ( -.  x R B  <->  -.  C R B ) )
2322rspcv 3155 . . . . . 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 4077 . . . . . . 7  |-  ( B  e.  A  ->  B  e.  { B ,  C } )
2625ad2antrl 726 . . . . . 6  |-  ( ( R  Fr  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  B  e.  { B ,  C } )
27 breq1 4397 . . . . . . . 8  |-  ( x  =  B  ->  (
x R C  <->  B R C ) )
2827notbid 292 . . . . . . 7  |-  ( x  =  B  ->  ( -.  x R C  <->  -.  B R C ) )
2928rspcv 3155 . . . . . 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 839 . . . 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 386 . 2  |-  ( ( R  Fr  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  ( -.  B R C  \/  -.  C R B ) )
34 ianor 486 . 2  |-  ( -.  ( B R C  /\  C R B )  <->  ( -.  B R C  \/  -.  C R B ) )
3533, 34sylibr 212 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 184    \/ wo 366    /\ wa 367    = wceq 1405    e. wcel 1842    =/= wne 2598   A.wral 2753   E.wrex 2754   _Vcvv 3058    C_ wss 3413   (/)c0 3737   {cpr 3973   class class class wbr 4394    Fr wfr 4778
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pr 4629
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-sbc 3277  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-op 3978  df-br 4395  df-fr 4781
This theorem is referenced by:  efrn2lp  4804  dfwe2  6598
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