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Theorem fr2nr 4703
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 4539 . . . . . . 7  |-  { B ,  C }  e.  _V
21a1i 11 . . . . . 6  |-  ( ( R  Fr  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  { B ,  C }  e.  _V )
3 simpl 457 . . . . . 6  |-  ( ( R  Fr  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  R  Fr  A )
4 prssi 4034 . . . . . . 7  |-  ( ( B  e.  A  /\  C  e.  A )  ->  { B ,  C }  C_  A )
54adantl 466 . . . . . 6  |-  ( ( R  Fr  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  { B ,  C }  C_  A
)
6 prnzg 4000 . . . . . . 7  |-  ( B  e.  A  ->  { B ,  C }  =/=  (/) )
76ad2antrl 727 . . . . . 6  |-  ( ( R  Fr  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  { B ,  C }  =/=  (/) )
8 fri 4687 . . . . . 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 1219 . . . . 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 4301 . . . . . . . . 9  |-  ( y  =  B  ->  (
x R y  <->  x R B ) )
1110notbid 294 . . . . . . . 8  |-  ( y  =  B  ->  ( -.  x R y  <->  -.  x R B ) )
1211ralbidv 2740 . . . . . . 7  |-  ( y  =  B  ->  ( A. x  e.  { B ,  C }  -.  x R y  <->  A. x  e.  { B ,  C }  -.  x R B ) )
13 breq2 4301 . . . . . . . . 9  |-  ( y  =  C  ->  (
x R y  <->  x R C ) )
1413notbid 294 . . . . . . . 8  |-  ( y  =  C  ->  ( -.  x R y  <->  -.  x R C ) )
1514ralbidv 2740 . . . . . . 7  |-  ( y  =  C  ->  ( A. x  e.  { B ,  C }  -.  x R y  <->  A. x  e.  { B ,  C }  -.  x R C ) )
1612, 15rexprg 3931 . . . . . 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 466 . . . . 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 3987 . . . . . . 7  |-  ( C  e.  A  ->  C  e.  { B ,  C } )
2019ad2antll 728 . . . . . 6  |-  ( ( R  Fr  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  C  e.  { B ,  C } )
21 breq1 4300 . . . . . . . 8  |-  ( x  =  C  ->  (
x R B  <->  C R B ) )
2221notbid 294 . . . . . . 7  |-  ( x  =  C  ->  ( -.  x R B  <->  -.  C R B ) )
2322rspcv 3074 . . . . . 6  |-  ( C  e.  { B ,  C }  ->  ( A. x  e.  { B ,  C }  -.  x R B  ->  -.  C R B ) )
2420, 23syl 16 . . . . 5  |-  ( ( R  Fr  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  ( A. x  e.  { B ,  C }  -.  x R B  ->  -.  C R B ) )
25 prid1g 3986 . . . . . . 7  |-  ( B  e.  A  ->  B  e.  { B ,  C } )
2625ad2antrl 727 . . . . . 6  |-  ( ( R  Fr  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  B  e.  { B ,  C } )
27 breq1 4300 . . . . . . . 8  |-  ( x  =  B  ->  (
x R C  <->  B R C ) )
2827notbid 294 . . . . . . 7  |-  ( x  =  B  ->  ( -.  x R C  <->  -.  B R C ) )
2928rspcv 3074 . . . . . 6  |-  ( B  e.  { B ,  C }  ->  ( A. x  e.  { B ,  C }  -.  x R C  ->  -.  B R C ) )
3026, 29syl 16 . . . . 5  |-  ( ( R  Fr  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  ( A. x  e.  { B ,  C }  -.  x R C  ->  -.  B R C ) )
3124, 30orim12d 834 . . . 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 388 . 2  |-  ( ( R  Fr  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  ( -.  B R C  \/  -.  C R B ) )
34 ianor 488 . 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 368    /\ wa 369    = wceq 1369    e. wcel 1756    =/= wne 2611   A.wral 2720   E.wrex 2721   _Vcvv 2977    C_ wss 3333   (/)c0 3642   {cpr 3884   class class class wbr 4297    Fr wfr 4681
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pr 4536
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-rab 2729  df-v 2979  df-sbc 3192  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-sn 3883  df-pr 3885  df-op 3889  df-br 4298  df-fr 4684
This theorem is referenced by:  efrn2lp  4707  dfwe2  6398
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