Metamath Proof Explorer


Theorem cossss

Description: Subclass theorem for the classes of cosets by A and B . (Contributed by Peter Mazsa, 11-Nov-2019)

Ref Expression
Assertion cossss A B A B

Proof

Step Hyp Ref Expression
1 ssbr A B x A y x B y
2 ssbr A B x A z x B z
3 1 2 anim12d A B x A y x A z x B y x B z
4 3 eximdv A B x x A y x A z x x B y x B z
5 4 ssopab2dv A B y z | x x A y x A z y z | x x B y x B z
6 df-coss A = y z | x x A y x A z
7 df-coss B = y z | x x B y x B z
8 5 6 7 3sstr4g A B A B