Description: Contraposition of negative in 'less than'. (Contributed by Mario Carneiro, 27-May-2016)
Ref | Expression | ||
---|---|---|---|
Hypotheses | leidd.1 | |- ( ph -> A e. RR ) |
|
ltnegd.2 | |- ( ph -> B e. RR ) |
||
ltnegcon1d.3 | |- ( ph -> -u A < B ) |
||
Assertion | ltnegcon1d | |- ( ph -> -u B < A ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | leidd.1 | |- ( ph -> A e. RR ) |
|
2 | ltnegd.2 | |- ( ph -> B e. RR ) |
|
3 | ltnegcon1d.3 | |- ( ph -> -u A < B ) |
|
4 | ltnegcon1 | |- ( ( A e. RR /\ B e. RR ) -> ( -u A < B <-> -u B < A ) ) |
|
5 | 1 2 4 | syl2anc | |- ( ph -> ( -u A < B <-> -u B < A ) ) |
6 | 3 5 | mpbid | |- ( ph -> -u B < A ) |