This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.
Metamath Proof Explorer
Description: Alternate definition of dominance. (Contributed by NM, 17-Jun-1998)
|
|
Ref |
Expression |
|
Assertion |
dfdom2 |
⊢ ≼ = ( ≺ ∪ ≈ ) |
Proof
| Step |
Hyp |
Ref |
Expression |
| 1 |
|
df-sdom |
⊢ ≺ = ( ≼ ∖ ≈ ) |
| 2 |
1
|
uneq2i |
⊢ ( ≈ ∪ ≺ ) = ( ≈ ∪ ( ≼ ∖ ≈ ) ) |
| 3 |
|
uncom |
⊢ ( ≈ ∪ ≺ ) = ( ≺ ∪ ≈ ) |
| 4 |
|
enssdom |
⊢ ≈ ⊆ ≼ |
| 5 |
|
undif |
⊢ ( ≈ ⊆ ≼ ↔ ( ≈ ∪ ( ≼ ∖ ≈ ) ) = ≼ ) |
| 6 |
4 5
|
mpbi |
⊢ ( ≈ ∪ ( ≼ ∖ ≈ ) ) = ≼ |
| 7 |
2 3 6
|
3eqtr3ri |
⊢ ≼ = ( ≺ ∪ ≈ ) |