Theorem brm 4134

Description: If two sets are in a binary relation, the relation is inhabited. (Contributed by Jim Kingdon, 31-Dec-2023.)

Assertion

Ref	Expression
brm	|- ( A R B -> E. x x e. R )

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

Proof of Theorem brm

Step	Hyp	Ref	Expression
1		df-br 4084	. 2 |- ( A R B <-> <. A , B >. e. R )
2		elex2 2816	. 2 |- ( <. A , B >. e. R -> E. x x e. R )
3	1, 2	sylbi 121	1 |- ( A R B -> E. x x e. R )

Syntax hints:   -> wi 4   E.wex 1538   e. wcel 2200   <.cop 3669   class class class wbr 4083

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-5 1493  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-v 2801  df-br 4084

This theorem is referenced by:  elfvm  5660