Description: Obsolete version of infsdomnn as of 7-Jan-2025. (Contributed by NM, 22-Nov-2004) (Revised by Mario Carneiro, 27-Apr-2015) (Proof modification is discouraged.) (New usage is discouraged.)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | infsdomnnOLD | |- ( ( _om ~<_ A /\ B e. _om ) -> B ~< A ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | reldom | |- Rel ~<_ |
|
| 2 | 1 | brrelex1i | |- ( _om ~<_ A -> _om e. _V ) |
| 3 | nnsdomg | |- ( ( _om e. _V /\ B e. _om ) -> B ~< _om ) |
|
| 4 | 2 3 | sylan | |- ( ( _om ~<_ A /\ B e. _om ) -> B ~< _om ) |
| 5 | simpl | |- ( ( _om ~<_ A /\ B e. _om ) -> _om ~<_ A ) |
|
| 6 | sdomdomtr | |- ( ( B ~< _om /\ _om ~<_ A ) -> B ~< A ) |
|
| 7 | 4 5 6 | syl2anc | |- ( ( _om ~<_ A /\ B e. _om ) -> B ~< A ) |