MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  frirr Structured version   Visualization version   Unicode version

Theorem frirr 5091
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 473 . . 3  |-  ( ( R  Fr  A  /\  B  e.  A )  ->  R  Fr  A )
2 snssi 4339 . . . 4  |-  ( B  e.  A  ->  { B }  C_  A )
32adantl 482 . . 3  |-  ( ( R  Fr  A  /\  B  e.  A )  ->  { B }  C_  A )
4 snnzg 4308 . . . 4  |-  ( B  e.  A  ->  { B }  =/=  (/) )
54adantl 482 . . 3  |-  ( ( R  Fr  A  /\  B  e.  A )  ->  { B }  =/=  (/) )
6 snex 4908 . . . 4  |-  { B }  e.  _V
76frc 5080 . . 3  |-  ( ( R  Fr  A  /\  { B }  C_  A  /\  { B }  =/=  (/) )  ->  E. y  e.  { B }  {
x  e.  { B }  |  x R
y }  =  (/) )
81, 3, 5, 7syl3anc 1326 . 2  |-  ( ( R  Fr  A  /\  B  e.  A )  ->  E. y  e.  { B }  { x  e.  { B }  |  x R y }  =  (/) )
9 rabeq0 3957 . . . . . 6  |-  ( { x  e.  { B }  |  x R
y }  =  (/)  <->  A. x  e.  { B }  -.  x R y )
10 breq2 4657 . . . . . . . 8  |-  ( y  =  B  ->  (
x R y  <->  x R B ) )
1110notbid 308 . . . . . . 7  |-  ( y  =  B  ->  ( -.  x R y  <->  -.  x R B ) )
1211ralbidv 2986 . . . . . 6  |-  ( y  =  B  ->  ( A. x  e.  { B }  -.  x R y  <->  A. x  e.  { B }  -.  x R B ) )
139, 12syl5bb 272 . . . . 5  |-  ( y  =  B  ->  ( { x  e.  { B }  |  x R
y }  =  (/)  <->  A. x  e.  { B }  -.  x R B ) )
1413rexsng 4219 . . . 4  |-  ( B  e.  A  ->  ( E. y  e.  { B }  { x  e.  { B }  |  x R y }  =  (/)  <->  A. x  e.  { B }  -.  x R B ) )
15 breq1 4656 . . . . . 6  |-  ( x  =  B  ->  (
x R B  <->  B R B ) )
1615notbid 308 . . . . 5  |-  ( x  =  B  ->  ( -.  x R B  <->  -.  B R B ) )
1716ralsng 4218 . . . 4  |-  ( B  e.  A  ->  ( A. x  e.  { B }  -.  x R B  <->  -.  B R B ) )
1814, 17bitrd 268 . . 3  |-  ( B  e.  A  ->  ( E. y  e.  { B }  { x  e.  { B }  |  x R y }  =  (/)  <->  -.  B R B ) )
1918adantl 482 . 2  |-  ( ( R  Fr  A  /\  B  e.  A )  ->  ( E. y  e. 
{ B }  {
x  e.  { B }  |  x R
y }  =  (/)  <->  -.  B R B ) )
208, 19mpbid 222 1  |-  ( ( R  Fr  A  /\  B  e.  A )  ->  -.  B R B )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 196    /\ wa 384    = wceq 1483    e. wcel 1990    =/= wne 2794   A.wral 2912   E.wrex 2913   {crab 2916    C_ wss 3574   (/)c0 3915   {csn 4177   class class class wbr 4653    Fr wfr 5070
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-fr 5073
This theorem is referenced by:  efrirr  5095  predfrirr  5709  dfwe2  6981  bnj1417  31109  efrunt  31590  ifr0  38654
  Copyright terms: Public domain W3C validator