Description: The converse of the singleton of the empty set is empty. (Contributed by Mario Carneiro, 30-Aug-2015)
Ref | Expression | ||
---|---|---|---|
Assertion | cnvsn0 | ⊢ ◡ { ∅ } = ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfdm4 | ⊢ dom { ∅ } = ran ◡ { ∅ } | |
2 | dmsn0 | ⊢ dom { ∅ } = ∅ | |
3 | 1 2 | eqtr3i | ⊢ ran ◡ { ∅ } = ∅ |
4 | relcnv | ⊢ Rel ◡ { ∅ } | |
5 | relrn0 | ⊢ ( Rel ◡ { ∅ } → ( ◡ { ∅ } = ∅ ↔ ran ◡ { ∅ } = ∅ ) ) | |
6 | 4 5 | ax-mp | ⊢ ( ◡ { ∅ } = ∅ ↔ ran ◡ { ∅ } = ∅ ) |
7 | 3 6 | mpbir | ⊢ ◡ { ∅ } = ∅ |