Theorem elopabr 5013

Description: Membership in a class abstraction of pairs, defined by a binary relation. (Contributed by AV, 16-Feb-2021.)

Assertion

Ref	Expression
elopabr	|- ( A e. { <. x , y >. | x R y } -> A e. R )

Distinct variable groups:   x, A, y   x, R, y

Proof of Theorem elopabr

Step	Hyp	Ref	Expression
1		elopab 4983	. 2 |- ( A e. { <. x , y >. | x R y } <-> E. x E. y ( A = <. x , y >. /\ x R y ) )
2		df-br 4654	. . . . . 6 |- ( x R y <-> <. x , y >. e. R )
3	2	biimpi 206	. . . . 5 |- ( x R y -> <. x , y >. e. R )
4		eleq1 2689	. . . . 5 |- ( A = <. x , y >. -> ( A e. R <-> <. x , y >. e. R ) )
5	3, 4	syl5ibr 236	. . . 4 |- ( A = <. x , y >. -> ( x R y -> A e. R ) )
6	5	imp 445	. . 3 |- ( ( A = <. x , y >. /\ x R y ) -> A e. R )
7	6	exlimivv 1860	. 2 |- ( E. x E. y ( A = <. x , y >. /\ x R y ) -> A e. R )
8	1, 7	sylbi 207	1 |- ( A e. { <. x , y >. | x R y } -> A e. R )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   <.cop 4183   class class class wbr 4653   {copab 4712

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713

This theorem is referenced by:  elopabran  5014