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Theorem frirr 4369
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 )
Dummy variables  x  y are mutually distinct and distinct from all other variables.

Proof of Theorem frirr
StepHypRef Expression
1 simpl 445 . . 3  |-  ( ( R  Fr  A  /\  B  e.  A )  ->  R  Fr  A )
2 simpr 449 . . . 4  |-  ( ( R  Fr  A  /\  B  e.  A )  ->  B  e.  A )
32snssd 3761 . . 3  |-  ( ( R  Fr  A  /\  B  e.  A )  ->  { B }  C_  A )
4 snnzg 3744 . . . 4  |-  ( B  e.  A  ->  { B }  =/=  (/) )
54adantl 454 . . 3  |-  ( ( R  Fr  A  /\  B  e.  A )  ->  { B }  =/=  (/) )
6 snex 4215 . . . 4  |-  { B }  e.  _V
76frc 4358 . . 3  |-  ( ( R  Fr  A  /\  { B }  C_  A  /\  { B }  =/=  (/) )  ->  E. y  e.  { B }  {
x  e.  { B }  |  x R
y }  =  (/) )
81, 3, 5, 7syl3anc 1184 . 2  |-  ( ( R  Fr  A  /\  B  e.  A )  ->  E. y  e.  { B }  { x  e.  { B }  |  x R y }  =  (/) )
9 rabeq0 3477 . . . . . 6  |-  ( { x  e.  { B }  |  x R
y }  =  (/)  <->  A. x  e.  { B }  -.  x R y )
10 breq2 4028 . . . . . . . 8  |-  ( y  =  B  ->  (
x R y  <->  x R B ) )
1110notbid 287 . . . . . . 7  |-  ( y  =  B  ->  ( -.  x R y  <->  -.  x R B ) )
1211ralbidv 2564 . . . . . 6  |-  ( y  =  B  ->  ( A. x  e.  { B }  -.  x R y  <->  A. x  e.  { B }  -.  x R B ) )
139, 12syl5bb 250 . . . . 5  |-  ( y  =  B  ->  ( { x  e.  { B }  |  x R
y }  =  (/)  <->  A. x  e.  { B }  -.  x R B ) )
1413rexsng 3674 . . . 4  |-  ( B  e.  A  ->  ( E. y  e.  { B }  { x  e.  { B }  |  x R y }  =  (/)  <->  A. x  e.  { B }  -.  x R B ) )
15 breq1 4027 . . . . . 6  |-  ( x  =  B  ->  (
x R B  <->  B R B ) )
1615notbid 287 . . . . 5  |-  ( x  =  B  ->  ( -.  x R B  <->  -.  B R B ) )
1716ralsng 3673 . . . 4  |-  ( B  e.  A  ->  ( A. x  e.  { B }  -.  x R B  <->  -.  B R B ) )
1814, 17bitrd 246 . . 3  |-  ( B  e.  A  ->  ( E. y  e.  { B }  { x  e.  { B }  |  x R y }  =  (/)  <->  -.  B R B ) )
1918adantl 454 . 2  |-  ( ( R  Fr  A  /\  B  e.  A )  ->  ( E. y  e. 
{ B }  {
x  e.  { B }  |  x R
y }  =  (/)  <->  -.  B R B ) )
208, 19mpbid 203 1  |-  ( ( R  Fr  A  /\  B  e.  A )  ->  -.  B R B )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    <-> wb 178    /\ wa 360    = wceq 1624    e. wcel 1685    =/= wne 2447   A.wral 2544   E.wrex 2545   {crab 2548    C_ wss 3153   (/)c0 3456   {csn 3641   class class class wbr 4024    Fr wfr 4348
This theorem is referenced by:  efrirr  4373  dfwe2  4572  efrunt  23463  predfrirr  23599  ifr0  27052  bnj1417  28338
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pr 4213
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-rab 2553  df-v 2791  df-sbc 2993  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-sn 3647  df-pr 3648  df-op 3650  df-br 4025  df-fr 4351
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