Metamath Proof Explorer

Theorem ssoprab2

Description: Equivalence of ordered pair abstraction subclass and implication. Compare ssopab2 . (Contributed by FL, 6-Nov-2013) (Proof shortened by Mario Carneiro, 11-Dec-2016)

Ref Expression
Assertion ssoprab2 
|- ( A. x A. y A. z ( ph -> ps ) -> { <. <. x , y >. , z >. | ph } C_ { <. <. x , y >. , z >. | ps } )

Proof

Step Hyp Ref Expression
1 id 
 |-  ( ( ph -> ps ) -> ( ph -> ps ) )
2 1 anim2d 
 |-  ( ( ph -> ps ) -> ( ( w = <. <. x , y >. , z >. /\ ph ) -> ( w = <. <. x , y >. , z >. /\ ps ) ) )
3 2 aleximi 
 |-  ( A. z ( ph -> ps ) -> ( E. z ( w = <. <. x , y >. , z >. /\ ph ) -> E. z ( w = <. <. x , y >. , z >. /\ ps ) ) )
4 3 aleximi 
 |-  ( A. y A. z ( ph -> ps ) -> ( E. y E. z ( w = <. <. x , y >. , z >. /\ ph ) -> E. y E. z ( w = <. <. x , y >. , z >. /\ ps ) ) )
5 4 aleximi 
 |-  ( A. x A. y A. z ( ph -> ps ) -> ( E. x E. y E. z ( w = <. <. x , y >. , z >. /\ ph ) -> E. x E. y E. z ( w = <. <. x , y >. , z >. /\ ps ) ) )
6 5 ss2abdv 
 |-  ( A. x A. y A. z ( ph -> ps ) -> { w | E. x E. y E. z ( w = <. <. x , y >. , z >. /\ ph ) } C_ { w | E. x E. y E. z ( w = <. <. x , y >. , z >. /\ ps ) } )
7 df-oprab 
 |-  { <. <. x , y >. , z >. | ph } = { w | E. x E. y E. z ( w = <. <. x , y >. , z >. /\ ph ) }
8 df-oprab 
 |-  { <. <. x , y >. , z >. | ps } = { w | E. x E. y E. z ( w = <. <. x , y >. , z >. /\ ps ) }
9 6 7 8 3sstr4g 
 |-  ( A. x A. y A. z ( ph -> ps ) -> { <. <. x , y >. , z >. | ph } C_ { <. <. x , y >. , z >. | ps } )