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Theorem frirr 4849
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 4165 . . 3  |-  ( ( R  Fr  A  /\  B  e.  A )  ->  { B }  C_  A )
4 snnzg 4137 . . . 4  |-  ( B  e.  A  ->  { B }  =/=  (/) )
54adantl 466 . . 3  |-  ( ( R  Fr  A  /\  B  e.  A )  ->  { B }  =/=  (/) )
6 snex 4681 . . . 4  |-  { B }  e.  _V
76frc 4838 . . 3  |-  ( ( R  Fr  A  /\  { B }  C_  A  /\  { B }  =/=  (/) )  ->  E. y  e.  { B }  {
x  e.  { B }  |  x R
y }  =  (/) )
81, 3, 5, 7syl3anc 1223 . 2  |-  ( ( R  Fr  A  /\  B  e.  A )  ->  E. y  e.  { B }  { x  e.  { B }  |  x R y }  =  (/) )
9 rabeq0 3800 . . . . . 6  |-  ( { x  e.  { B }  |  x R
y }  =  (/)  <->  A. x  e.  { B }  -.  x R y )
10 breq2 4444 . . . . . . . 8  |-  ( y  =  B  ->  (
x R y  <->  x R B ) )
1110notbid 294 . . . . . . 7  |-  ( y  =  B  ->  ( -.  x R y  <->  -.  x R B ) )
1211ralbidv 2896 . . . . . 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 4056 . . . 4  |-  ( B  e.  A  ->  ( E. y  e.  { B }  { x  e.  { B }  |  x R y }  =  (/)  <->  A. x  e.  { B }  -.  x R B ) )
15 breq1 4443 . . . . . 6  |-  ( x  =  B  ->  (
x R B  <->  B R B ) )
1615notbid 294 . . . . 5  |-  ( x  =  B  ->  ( -.  x R B  <->  -.  B R B ) )
1716ralsng 4055 . . . 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 1374    e. wcel 1762    =/= wne 2655   A.wral 2807   E.wrex 2808   {crab 2811    C_ wss 3469   (/)c0 3778   {csn 4020   class class class wbr 4440    Fr wfr 4828
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pr 4679
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-sbc 3325  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-sn 4021  df-pr 4023  df-op 4027  df-br 4441  df-fr 4831
This theorem is referenced by:  efrirr  4853  dfwe2  6588  efrunt  28546  predfrirr  28841  ifr0  30892  bnj1417  33051
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