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Theorem fr2nr 4343
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
StepHypRef Expression
1 prex 4189 . . . . . . 7  |-  { B ,  C }  e.  _V
21a1i 12 . . . . . 6  |-  ( ( R  Fr  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  { B ,  C }  e.  _V )
3 simpl 445 . . . . . 6  |-  ( ( R  Fr  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  R  Fr  A )
4 prssi 3745 . . . . . . 7  |-  ( ( B  e.  A  /\  C  e.  A )  ->  { B ,  C }  C_  A )
54adantl 454 . . . . . 6  |-  ( ( R  Fr  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  { B ,  C }  C_  A
)
6 prnzg 3720 . . . . . . 7  |-  ( B  e.  A  ->  { B ,  C }  =/=  (/) )
76ad2antrl 711 . . . . . 6  |-  ( ( R  Fr  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  { B ,  C }  =/=  (/) )
8 fri 4327 . . . . . 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 1188 . . . . 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 4001 . . . . . . . . 9  |-  ( y  =  B  ->  (
x R y  <->  x R B ) )
1110notbid 287 . . . . . . . 8  |-  ( y  =  B  ->  ( -.  x R y  <->  -.  x R B ) )
1211ralbidv 2538 . . . . . . 7  |-  ( y  =  B  ->  ( A. x  e.  { B ,  C }  -.  x R y  <->  A. x  e.  { B ,  C }  -.  x R B ) )
13 breq2 4001 . . . . . . . . 9  |-  ( y  =  C  ->  (
x R y  <->  x R C ) )
1413notbid 287 . . . . . . . 8  |-  ( y  =  C  ->  ( -.  x R y  <->  -.  x R C ) )
1514ralbidv 2538 . . . . . . 7  |-  ( y  =  C  ->  ( A. x  e.  { B ,  C }  -.  x R y  <->  A. x  e.  { B ,  C }  -.  x R C ) )
1612, 15rexprg 3657 . . . . . 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 454 . . . . 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 203 . . . 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 3707 . . . . . . 7  |-  ( C  e.  A  ->  C  e.  { B ,  C } )
2019ad2antll 712 . . . . . 6  |-  ( ( R  Fr  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  C  e.  { B ,  C } )
21 breq1 4000 . . . . . . . 8  |-  ( x  =  C  ->  (
x R B  <->  C R B ) )
2221notbid 287 . . . . . . 7  |-  ( x  =  C  ->  ( -.  x R B  <->  -.  C R B ) )
2322rcla4v 2855 . . . . . 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 3706 . . . . . . 7  |-  ( B  e.  A  ->  B  e.  { B ,  C } )
2625ad2antrl 711 . . . . . 6  |-  ( ( R  Fr  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  B  e.  { B ,  C } )
27 breq1 4000 . . . . . . . 8  |-  ( x  =  B  ->  (
x R C  <->  B R C ) )
2827notbid 287 . . . . . . 7  |-  ( x  =  B  ->  ( -.  x R C  <->  -.  B R C ) )
2928rcla4v 2855 . . . . . 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 814 . . . 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 16 . . 3  |-  ( ( R  Fr  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  ( -.  C R B  \/  -.  B R C ) )
3332orcomd 379 . 2  |-  ( ( R  Fr  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  ( -.  B R C  \/  -.  C R B ) )
34 ianor 476 . 2  |-  ( -.  ( B R C  /\  C R B )  <->  ( -.  B R C  \/  -.  C R B ) )
3533, 34sylibr 205 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 5    -> wi 6    <-> wb 178    \/ wo 359    /\ wa 360    = wceq 1619    e. wcel 1621    =/= wne 2421   A.wral 2518   E.wrex 2519   _Vcvv 2763    C_ wss 3127   (/)c0 3430   {cpr 3615   class class class wbr 3997    Fr wfr 4321
This theorem is referenced by:  efrn2lp  4347  dfwe2  4545
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-sep 4115  ax-nul 4123  ax-pr 4186
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-ral 2523  df-rex 2524  df-rab 2527  df-v 2765  df-sbc 2967  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-nul 3431  df-if 3540  df-sn 3620  df-pr 3621  df-op 3623  df-br 3998  df-fr 4324
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