Theorem elopab 4378

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

Assertion

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

Distinct variable groups:   x, A   y, A

Allowed substitution hints:   ph( x, y)

Proof of Theorem elopab

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 2827	. 2 |- ( A e. { <. x , y >. | ph } -> A e. _V )
2		vex 2818	. . . . . 6 |- x e. _V
3		vex 2818	. . . . . 6 |- y e. _V
4	2, 3	opex 4347	. . . . 5 |- <. x , y >. e. _V
5		eleq1 2297	. . . . 5 |- ( A = <. x , y >. -> ( A e. _V <-> <. x , y >. e. _V ) )
6	4, 5	mpbiri 168	. . . 4 |- ( A = <. x , y >. -> A e. _V )
7	6	adantr 276	. . 3 |- ( ( A = <. x , y >. /\ ph ) -> A e. _V )
8	7	exlimivv 1948	. 2 |- ( E. x E. y ( A = <. x , y >. /\ ph ) -> A e. _V )
9		eqeq1 2241	. . . . 5 |- ( z = A -> ( z = <. x , y >. <-> A = <. x , y >. ) )
10	9	anbi1d 465	. . . 4 |- ( z = A -> ( ( z = <. x , y >. /\ ph ) <-> ( A = <. x , y >. /\ ph ) ) )
11	10	2exbidv 1917	. . 3 |- ( z = A -> ( E. x E. y ( z = <. x , y >. /\ ph ) <-> E. x E. y ( A = <. x , y >. /\ ph ) ) )
12		df-opab 4174	. . 3 |- { <. x , y >. | ph } = { z | E. x E. y ( z = <. x , y >. /\ ph ) }
13	11, 12	elab2g 2966	. 2 |- ( A e. _V -> ( A e. { <. x , y >. | ph } <-> E. x E. y ( A = <. x , y >. /\ ph ) ) )
14	1, 8, 13	pm5.21nii 712	1 |- ( A e. { <. x , y >. | ph } <-> E. x E. y ( A = <. x , y >. /\ ph ) )

Syntax hints:   /\ wa 104   <-> wb 105   = wceq 1398   E.wex 1541   e. wcel 2205   _Vcvv 2815   <.cop 3694   {copab 4172

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4230  ax-pow 4289  ax-pr 4324

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-v 2817  df-un 3217  df-in 3219  df-ss 3226  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-opab 4174

This theorem is referenced by:  opelopabsbALT  4379  opelopabsb  4380  opelopabt  4382  opelopabga  4383  opabm  4401  iunopab  4402  epelg  4413  elxp  4768  elco  4923  elcnv  4934  dfmpt3  5483  0neqopab  6100  brabvv  6101  opabex3d  6316  opabex3  6317  griedg0ssusgr  16263