Metamath Proof Explorer

Theorem elopab

Description: Membership in a class abstraction of ordered pairs. (Contributed by NM, 24-Mar-1998)

Ref Expression
Assertion elopab 
|- ( A e. { <. x , y >. | ph } <-> E. x E. y ( A = <. x , y >. /\ ph ) )

Proof

Step Hyp Ref Expression
1 elex 
 |-  ( A e. { <. x , y >. | ph } -> A e. _V )
2 opex 
 |-  <. x , y >. e. _V
3 eleq1 
 |-  ( A = <. x , y >. -> ( A e. _V <-> <. x , y >. e. _V ) )
4 2 3 mpbiri 
 |-  ( A = <. x , y >. -> A e. _V )
5 4 adantr 
 |-  ( ( A = <. x , y >. /\ ph ) -> A e. _V )
6 5 exlimivv 
 |-  ( E. x E. y ( A = <. x , y >. /\ ph ) -> A e. _V )
7 eqeq1 
 |-  ( z = w -> ( z = <. x , y >. <-> w = <. x , y >. ) )
8 7 anbi1d 
 |-  ( z = w -> ( ( z = <. x , y >. /\ ph ) <-> ( w = <. x , y >. /\ ph ) ) )
9 8 2exbidv 
 |-  ( z = w -> ( E. x E. y ( z = <. x , y >. /\ ph ) <-> E. x E. y ( w = <. x , y >. /\ ph ) ) )
10 eqeq1 
 |-  ( w = A -> ( w = <. x , y >. <-> A = <. x , y >. ) )
11 10 anbi1d 
 |-  ( w = A -> ( ( w = <. x , y >. /\ ph ) <-> ( A = <. x , y >. /\ ph ) ) )
12 11 2exbidv 
 |-  ( w = A -> ( E. x E. y ( w = <. x , y >. /\ ph ) <-> E. x E. y ( A = <. x , y >. /\ ph ) ) )
13 df-opab 
 |-  { <. x , y >. | ph } = { z | E. x E. y ( z = <. x , y >. /\ ph ) }
14 9 12 13 elab2gw 
 |-  ( A e. _V -> ( A e. { <. x , y >. | ph } <-> E. x E. y ( A = <. x , y >. /\ ph ) ) )
15 1 6 14 pm5.21nii 
 |-  ( A e. { <. x , y >. | ph } <-> E. x E. y ( A = <. x , y >. /\ ph ) )