Description: Union is smaller than Hilbert lattice join. (Contributed by NM, 11-Jun-2004) (Revised by Mario Carneiro, 15-May-2014) (New usage is discouraged.)
| Ref | Expression | ||
|---|---|---|---|
| Hypotheses | shincl.1 | |- A e. SH |
|
| shincl.2 | |- B e. SH |
||
| Assertion | shunssji | |- ( A u. B ) C_ ( A vH B ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | shincl.1 | |- A e. SH |
|
| 2 | shincl.2 | |- B e. SH |
|
| 3 | 1 | shssii | |- A C_ ~H |
| 4 | 2 | shssii | |- B C_ ~H |
| 5 | 3 4 | unssi | |- ( A u. B ) C_ ~H |
| 6 | ococss | |- ( ( A u. B ) C_ ~H -> ( A u. B ) C_ ( _|_ ` ( _|_ ` ( A u. B ) ) ) ) |
|
| 7 | 5 6 | ax-mp | |- ( A u. B ) C_ ( _|_ ` ( _|_ ` ( A u. B ) ) ) |
| 8 | shjval | |- ( ( A e. SH /\ B e. SH ) -> ( A vH B ) = ( _|_ ` ( _|_ ` ( A u. B ) ) ) ) |
|
| 9 | 1 2 8 | mp2an | |- ( A vH B ) = ( _|_ ` ( _|_ ` ( A u. B ) ) ) |
| 10 | 7 9 | sseqtrri | |- ( A u. B ) C_ ( A vH B ) |