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Metamath Proof Explorer
Description: Alternate definition of strict dominance. Compare Definition 3 of
Suppes p. 97. (Contributed by NM, 31-Mar-1998)
|
|
Ref |
Expression |
|
Assertion |
dfsdom2 |
⊢ ≺ = ( ≼ ∖ ◡ ≼ ) |
Proof
| Step |
Hyp |
Ref |
Expression |
| 1 |
|
df-sdom |
⊢ ≺ = ( ≼ ∖ ≈ ) |
| 2 |
|
sbthcl |
⊢ ≈ = ( ≼ ∩ ◡ ≼ ) |
| 3 |
2
|
difeq2i |
⊢ ( ≼ ∖ ≈ ) = ( ≼ ∖ ( ≼ ∩ ◡ ≼ ) ) |
| 4 |
|
difin |
⊢ ( ≼ ∖ ( ≼ ∩ ◡ ≼ ) ) = ( ≼ ∖ ◡ ≼ ) |
| 5 |
1 3 4
|
3eqtri |
⊢ ≺ = ( ≼ ∖ ◡ ≼ ) |