Theorem brm 4095

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 4046	. 2 |- ( A R B <-> <. A , B >. e. R )
2		elex2 2788	. 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 1515   e. wcel 2176   <.cop 3636   class class class wbr 4045

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 1470  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-v 2774  df-br 4046

This theorem is referenced by:  elfvm  5611