Metamath Proof Explorer


Theorem occon

Description: Contraposition law for orthogonal complement. (Contributed by NM, 8-Aug-2000) (New usage is discouraged.)

Ref Expression
Assertion occon A B A B B A

Proof

Step Hyp Ref Expression
1 ssralv A B y B x ih y = 0 y A x ih y = 0
2 1 ralrimivw A B x y B x ih y = 0 y A x ih y = 0
3 ss2rab x | y B x ih y = 0 x | y A x ih y = 0 x y B x ih y = 0 y A x ih y = 0
4 2 3 sylibr A B x | y B x ih y = 0 x | y A x ih y = 0
5 4 adantl A B A B x | y B x ih y = 0 x | y A x ih y = 0
6 ocval B B = x | y B x ih y = 0
7 6 ad2antlr A B A B B = x | y B x ih y = 0
8 ocval A A = x | y A x ih y = 0
9 8 ad2antrr A B A B A = x | y A x ih y = 0
10 5 7 9 3sstr4d A B A B B A
11 10 ex A B A B B A