This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.
Metamath Proof Explorer
Theorem tc0
Description: The transitive closure of the empty set. (Contributed by Mario Carneiro, 4-Jun-2015)
|
|
Ref |
Expression |
|
Assertion |
tc0 |
⊢ ( TC ‘ ∅ ) = ∅ |
Proof
| Step |
Hyp |
Ref |
Expression |
| 1 |
|
ssid |
⊢ ∅ ⊆ ∅ |
| 2 |
|
tr0 |
⊢ Tr ∅ |
| 3 |
|
0ex |
⊢ ∅ ∈ V |
| 4 |
|
tcmin |
⊢ ( ∅ ∈ V → ( ( ∅ ⊆ ∅ ∧ Tr ∅ ) → ( TC ‘ ∅ ) ⊆ ∅ ) ) |
| 5 |
3 4
|
ax-mp |
⊢ ( ( ∅ ⊆ ∅ ∧ Tr ∅ ) → ( TC ‘ ∅ ) ⊆ ∅ ) |
| 6 |
1 2 5
|
mp2an |
⊢ ( TC ‘ ∅ ) ⊆ ∅ |
| 7 |
|
0ss |
⊢ ∅ ⊆ ( TC ‘ ∅ ) |
| 8 |
6 7
|
eqssi |
⊢ ( TC ‘ ∅ ) = ∅ |