Description: Any non-zero ordinal product is greater-than-or-equal to the term on the left. Lemma 3.11 of Schloeder p. 8. See omword1 . (Contributed by RP, 29-Jan-2025)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | omge1 | |- ( ( A e. On /\ B e. On /\ B =/= (/) ) -> A C_ ( A .o B ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | 3simpa | |- ( ( A e. On /\ B e. On /\ B =/= (/) ) -> ( A e. On /\ B e. On ) ) |
|
| 2 | on0eln0 | |- ( B e. On -> ( (/) e. B <-> B =/= (/) ) ) |
|
| 3 | 2 | biimpar | |- ( ( B e. On /\ B =/= (/) ) -> (/) e. B ) |
| 4 | 3 | 3adant1 | |- ( ( A e. On /\ B e. On /\ B =/= (/) ) -> (/) e. B ) |
| 5 | omword1 | |- ( ( ( A e. On /\ B e. On ) /\ (/) e. B ) -> A C_ ( A .o B ) ) |
|
| 6 | 1 4 5 | syl2anc | |- ( ( A e. On /\ B e. On /\ B =/= (/) ) -> A C_ ( A .o B ) ) |