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Theorem frirr 4806
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 457 . . 3  |-  ( ( R  Fr  A  /\  B  e.  A )  ->  R  Fr  A )
2 simpr 461 . . . 4  |-  ( ( R  Fr  A  /\  B  e.  A )  ->  B  e.  A )
32snssd 4127 . . 3  |-  ( ( R  Fr  A  /\  B  e.  A )  ->  { B }  C_  A )
4 snnzg 4101 . . . 4  |-  ( B  e.  A  ->  { B }  =/=  (/) )
54adantl 466 . . 3  |-  ( ( R  Fr  A  /\  B  e.  A )  ->  { B }  =/=  (/) )
6 snex 4642 . . . 4  |-  { B }  e.  _V
76frc 4795 . . 3  |-  ( ( R  Fr  A  /\  { B }  C_  A  /\  { B }  =/=  (/) )  ->  E. y  e.  { B }  {
x  e.  { B }  |  x R
y }  =  (/) )
81, 3, 5, 7syl3anc 1219 . 2  |-  ( ( R  Fr  A  /\  B  e.  A )  ->  E. y  e.  { B }  { x  e.  { B }  |  x R y }  =  (/) )
9 rabeq0 3768 . . . . . 6  |-  ( { x  e.  { B }  |  x R
y }  =  (/)  <->  A. x  e.  { B }  -.  x R y )
10 breq2 4405 . . . . . . . 8  |-  ( y  =  B  ->  (
x R y  <->  x R B ) )
1110notbid 294 . . . . . . 7  |-  ( y  =  B  ->  ( -.  x R y  <->  -.  x R B ) )
1211ralbidv 2846 . . . . . 6  |-  ( y  =  B  ->  ( A. x  e.  { B }  -.  x R y  <->  A. x  e.  { B }  -.  x R B ) )
139, 12syl5bb 257 . . . . 5  |-  ( y  =  B  ->  ( { x  e.  { B }  |  x R
y }  =  (/)  <->  A. x  e.  { B }  -.  x R B ) )
1413rexsng 4022 . . . 4  |-  ( B  e.  A  ->  ( E. y  e.  { B }  { x  e.  { B }  |  x R y }  =  (/)  <->  A. x  e.  { B }  -.  x R B ) )
15 breq1 4404 . . . . . 6  |-  ( x  =  B  ->  (
x R B  <->  B R B ) )
1615notbid 294 . . . . 5  |-  ( x  =  B  ->  ( -.  x R B  <->  -.  B R B ) )
1716ralsng 4021 . . . 4  |-  ( B  e.  A  ->  ( A. x  e.  { B }  -.  x R B  <->  -.  B R B ) )
1814, 17bitrd 253 . . 3  |-  ( B  e.  A  ->  ( E. y  e.  { B }  { x  e.  { B }  |  x R y }  =  (/)  <->  -.  B R B ) )
1918adantl 466 . 2  |-  ( ( R  Fr  A  /\  B  e.  A )  ->  ( E. y  e. 
{ B }  {
x  e.  { B }  |  x R
y }  =  (/)  <->  -.  B R B ) )
208, 19mpbid 210 1  |-  ( ( R  Fr  A  /\  B  e.  A )  ->  -.  B R B )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758    =/= wne 2648   A.wral 2799   E.wrex 2800   {crab 2803    C_ wss 3437   (/)c0 3746   {csn 3986   class class class wbr 4401    Fr wfr 4785
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4522  ax-nul 4530  ax-pr 4640
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-sn 3987  df-pr 3989  df-op 3993  df-br 4402  df-fr 4788
This theorem is referenced by:  efrirr  4810  dfwe2  6504  efrunt  27509  predfrirr  27804  ifr0  29855  bnj1417  32365
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