Description: The rank of the union of two sets. Theorem 15.17(iii) of Monk1 p. 112. (Contributed by NM, 26-Nov-2003) (Revised by Mario Carneiro, 17-Nov-2014)
| Ref | Expression | ||
|---|---|---|---|
| Hypotheses | ranksn.1 | |- A e. _V |
|
| rankun.2 | |- B e. _V |
||
| Assertion | rankun | |- ( rank ` ( A u. B ) ) = ( ( rank ` A ) u. ( rank ` B ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ranksn.1 | |- A e. _V |
|
| 2 | rankun.2 | |- B e. _V |
|
| 3 | unir1 | |- U. ( R1 " On ) = _V |
|
| 4 | 1 3 | eleqtrri | |- A e. U. ( R1 " On ) |
| 5 | 2 3 | eleqtrri | |- B e. U. ( R1 " On ) |
| 6 | rankunb | |- ( ( A e. U. ( R1 " On ) /\ B e. U. ( R1 " On ) ) -> ( rank ` ( A u. B ) ) = ( ( rank ` A ) u. ( rank ` B ) ) ) |
|
| 7 | 4 5 6 | mp2an | |- ( rank ` ( A u. B ) ) = ( ( rank ` A ) u. ( rank ` B ) ) |