Description: The subset relation is inherited by the rank function. Exercise 1 of TakeutiZaring p. 80. (Contributed by NM, 25-Nov-2003) (Revised by Mario Carneiro, 17-Nov-2014)
| Ref | Expression | ||
|---|---|---|---|
| Hypothesis | rankss.1 | |- B e. _V |
|
| Assertion | rankss | |- ( A C_ B -> ( rank ` A ) C_ ( rank ` B ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | rankss.1 | |- B e. _V |
|
| 2 | unir1 | |- U. ( R1 " On ) = _V |
|
| 3 | 1 2 | eleqtrri | |- B e. U. ( R1 " On ) |
| 4 | rankssb | |- ( B e. U. ( R1 " On ) -> ( A C_ B -> ( rank ` A ) C_ ( rank ` B ) ) ) |
|
| 5 | 3 4 | ax-mp | |- ( A C_ B -> ( rank ` A ) C_ ( rank ` B ) ) |