Description: No subspace is smaller than the zero subspace. (Contributed by NM, 24-Nov-2004) (New usage is discouraged.)
Ref | Expression | ||
---|---|---|---|
Assertion | shle0 | |- ( A e. SH -> ( A C_ 0H <-> A = 0H ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sh0le | |- ( A e. SH -> 0H C_ A ) |
|
2 | 1 | biantrud | |- ( A e. SH -> ( A C_ 0H <-> ( A C_ 0H /\ 0H C_ A ) ) ) |
3 | eqss | |- ( A = 0H <-> ( A C_ 0H /\ 0H C_ A ) ) |
|
4 | 2 3 | bitr4di | |- ( A e. SH -> ( A C_ 0H <-> A = 0H ) ) |