Theorem elopabr 4370

Description: Membership in an ordered-pair class abstraction defined by a binary relation. (Contributed by AV, 16-Feb-2021.) (Proof shortened by SN, 11-Dec-2024.)

Assertion

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

Distinct variable groups:   x, R   y, R

Allowed substitution hints:   A( x, y)

Proof of Theorem elopabr

Step	Hyp	Ref	Expression
1		opabss 4147	. 2 |- { <. x , y >. | x R y } C_ R
2	1	sseli 3220	1 |- ( A e. { <. x , y >. | x R y } -> A e. R )

Syntax hints:   -> wi 4   e. wcel 2200   class class class wbr 4082   {copab 4143

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-in 3203  df-ss 3210  df-br 4083  df-opab 4145

This theorem is referenced by: (None)