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Metamath Proof Explorer

Theorem coss0

Description: Cosets by the empty set are the empty set. (Contributed by Peter Mazsa, 22-Oct-2019)

Ref Expression
Assertion coss0 
|- ,~ (/) = (/)

Proof

Step Hyp Ref Expression
1 dfcoss2 
 |-  ,~ (/) = { <. y , z >. | E. x ( y e. [ x ] (/) /\ z e. [ x ] (/) ) }
2 ec0 
 |-  [ x ] (/) = (/)
3 2 eleq2i 
 |-  ( y e. [ x ] (/) <-> y e. (/) )
4 2 eleq2i 
 |-  ( z e. [ x ] (/) <-> z e. (/) )
5 3 4 anbi12i 
 |-  ( ( y e. [ x ] (/) /\ z e. [ x ] (/) ) <-> ( y e. (/) /\ z e. (/) ) )
6 5 exbii 
 |-  ( E. x ( y e. [ x ] (/) /\ z e. [ x ] (/) ) <-> E. x ( y e. (/) /\ z e. (/) ) )
7 19.9v 
 |-  ( E. x ( y e. (/) /\ z e. (/) ) <-> ( y e. (/) /\ z e. (/) ) )
8 6 7 bitri 
 |-  ( E. x ( y e. [ x ] (/) /\ z e. [ x ] (/) ) <-> ( y e. (/) /\ z e. (/) ) )
9 8 opabbii 
 |-  { <. y , z >. | E. x ( y e. [ x ] (/) /\ z e. [ x ] (/) ) } = { <. y , z >. | ( y e. (/) /\ z e. (/) ) }
10 prnzg 
 |-  ( y e. _V -> { y , z } =/= (/) )
11 10 elv 
 |-  { y , z } =/= (/)
12 ss0b 
 |-  ( { y , z } C_ (/) <-> { y , z } = (/) )
13 11 12 nemtbir 
 |-  -. { y , z } C_ (/)
14 prssg 
 |-  ( ( y e. _V /\ z e. _V ) -> ( ( y e. (/) /\ z e. (/) ) <-> { y , z } C_ (/) ) )
15 14 el2v 
 |-  ( ( y e. (/) /\ z e. (/) ) <-> { y , z } C_ (/) )
16 13 15 mtbir 
 |-  -. ( y e. (/) /\ z e. (/) )
17 16 opabf 
 |-  { <. y , z >. | ( y e. (/) /\ z e. (/) ) } = (/)
18 1 9 17 3eqtri 
 |-  ,~ (/) = (/)