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Theorem frirr 4833
Description: A well-founded relation is irreflexive. Special case of Proposition 6.23 of [TakeutiZaring] p. 30. (Contributed by NM, 2-Jan-1994.) (Revised by Mario Carneiro, 22-Jun-2015.)
Assertion
Ref Expression
frirr  |-  ( ( R  Fr  A  /\  B  e.  A )  ->  -.  B R B )

Proof of Theorem frirr
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl 463 . . 3  |-  ( ( R  Fr  A  /\  B  e.  A )  ->  R  Fr  A )
2 simpr 467 . . . 4  |-  ( ( R  Fr  A  /\  B  e.  A )  ->  B  e.  A )
32snssd 4130 . . 3  |-  ( ( R  Fr  A  /\  B  e.  A )  ->  { B }  C_  A )
4 snnzg 4102 . . . 4  |-  ( B  e.  A  ->  { B }  =/=  (/) )
54adantl 472 . . 3  |-  ( ( R  Fr  A  /\  B  e.  A )  ->  { B }  =/=  (/) )
6 snex 4658 . . . 4  |-  { B }  e.  _V
76frc 4822 . . 3  |-  ( ( R  Fr  A  /\  { B }  C_  A  /\  { B }  =/=  (/) )  ->  E. y  e.  { B }  {
x  e.  { B }  |  x R
y }  =  (/) )
81, 3, 5, 7syl3anc 1276 . 2  |-  ( ( R  Fr  A  /\  B  e.  A )  ->  E. y  e.  { B }  { x  e.  { B }  |  x R y }  =  (/) )
9 rabeq0 3766 . . . . . 6  |-  ( { x  e.  { B }  |  x R
y }  =  (/)  <->  A. x  e.  { B }  -.  x R y )
10 breq2 4422 . . . . . . . 8  |-  ( y  =  B  ->  (
x R y  <->  x R B ) )
1110notbid 300 . . . . . . 7  |-  ( y  =  B  ->  ( -.  x R y  <->  -.  x R B ) )
1211ralbidv 2839 . . . . . 6  |-  ( y  =  B  ->  ( A. x  e.  { B }  -.  x R y  <->  A. x  e.  { B }  -.  x R B ) )
139, 12syl5bb 265 . . . . 5  |-  ( y  =  B  ->  ( { x  e.  { B }  |  x R
y }  =  (/)  <->  A. x  e.  { B }  -.  x R B ) )
1413rexsng 4019 . . . 4  |-  ( B  e.  A  ->  ( E. y  e.  { B }  { x  e.  { B }  |  x R y }  =  (/)  <->  A. x  e.  { B }  -.  x R B ) )
15 breq1 4421 . . . . . 6  |-  ( x  =  B  ->  (
x R B  <->  B R B ) )
1615notbid 300 . . . . 5  |-  ( x  =  B  ->  ( -.  x R B  <->  -.  B R B ) )
1716ralsng 4018 . . . 4  |-  ( B  e.  A  ->  ( A. x  e.  { B }  -.  x R B  <->  -.  B R B ) )
1814, 17bitrd 261 . . 3  |-  ( B  e.  A  ->  ( E. y  e.  { B }  { x  e.  { B }  |  x R y }  =  (/)  <->  -.  B R B ) )
1918adantl 472 . 2  |-  ( ( R  Fr  A  /\  B  e.  A )  ->  ( E. y  e. 
{ B }  {
x  e.  { B }  |  x R
y }  =  (/)  <->  -.  B R B ) )
208, 19mpbid 215 1  |-  ( ( R  Fr  A  /\  B  e.  A )  ->  -.  B R B )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 189    /\ wa 375    = wceq 1455    e. wcel 1898    =/= wne 2633   A.wral 2749   E.wrex 2750   {crab 2753    C_ wss 3416   (/)c0 3743   {csn 3980   class class class wbr 4418    Fr wfr 4812
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-sep 4541  ax-nul 4550  ax-pr 4656
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-ral 2754  df-rex 2755  df-rab 2758  df-v 3059  df-sbc 3280  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3744  df-if 3894  df-sn 3981  df-pr 3983  df-op 3987  df-br 4419  df-fr 4815
This theorem is referenced by:  efrirr  4837  predfrirr  5432  dfwe2  6640  bnj1417  29900  efrunt  30390  ifr0  36848
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