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Metamath Proof Explorer

Theorem frirr

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)

Ref Expression
Assertion frirr 
|- ( ( R Fr A /\ B e. A ) -> -. B R B )

Proof

Step Hyp Ref Expression
1 simpl 
 |-  ( ( R Fr A /\ B e. A ) -> R Fr A )
2 snssi 
 |-  ( B e. A -> { B } C_ A )
3 2 adantl 
 |-  ( ( R Fr A /\ B e. A ) -> { B } C_ A )
4 snnzg 
 |-  ( B e. A -> { B } =/= (/) )
5 4 adantl 
 |-  ( ( R Fr A /\ B e. A ) -> { B } =/= (/) )
6 snex 
 |-  { B } e. _V
7 6 frc 
 |-  ( ( R Fr A /\ { B } C_ A /\ { B } =/= (/) ) -> E. y e. { B } { x e. { B } | x R y } = (/) )
8 1 3 5 7 syl3anc 
 |-  ( ( R Fr A /\ B e. A ) -> E. y e. { B } { x e. { B } | x R y } = (/) )
9 breq1 
 |-  ( x = z -> ( x R y <-> z R y ) )
10 9 rabeq0w 
 |-  ( { x e. { B } | x R y } = (/) <-> A. z e. { B } -. z R y )
11 breq2 
 |-  ( y = B -> ( z R y <-> z R B ) )
12 11 notbid 
 |-  ( y = B -> ( -. z R y <-> -. z R B ) )
13 12 ralbidv 
 |-  ( y = B -> ( A. z e. { B } -. z R y <-> A. z e. { B } -. z R B ) )
14 10 13 bitrid 
 |-  ( y = B -> ( { x e. { B } | x R y } = (/) <-> A. z e. { B } -. z R B ) )
15 14 rexsng 
 |-  ( B e. A -> ( E. y e. { B } { x e. { B } | x R y } = (/) <-> A. z e. { B } -. z R B ) )
16 breq1 
 |-  ( z = B -> ( z R B <-> B R B ) )
17 16 notbid 
 |-  ( z = B -> ( -. z R B <-> -. B R B ) )
18 17 ralsng 
 |-  ( B e. A -> ( A. z e. { B } -. z R B <-> -. B R B ) )
19 15 18 bitrd 
 |-  ( B e. A -> ( E. y e. { B } { x e. { B } | x R y } = (/) <-> -. B R B ) )
20 19 adantl 
 |-  ( ( R Fr A /\ B e. A ) -> ( E. y e. { B } { x e. { B } | x R y } = (/) <-> -. B R B ) )
21 8 20 mpbid 
 |-  ( ( R Fr A /\ B e. A ) -> -. B R B )