Theorem elopabr 4383

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 4158	. 2 |- { <. x , y >. | x R y } C_ R
2	1	sseli 3224	1 |- ( A e. { <. x , y >. | x R y } -> A e. R )

Syntax hints:   -> wi 4   e. wcel 2202   class class class wbr 4093   {copab 4154

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-ext 2213

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-in 3207  df-ss 3214  df-br 4094  df-opab 4156

This theorem is referenced by: (None)