Theorem brm 4032

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 3983	. 2 |- ( A R B <-> <. A , B >. e. R )
2		elex2 2742	. 2 |- ( <. A , B >. e. R -> E. x x e. R )
3	1, 2	sylbi 120	1 |- ( A R B -> E. x x e. R )

Syntax hints:   -> wi 4   E.wex 1480   e. wcel 2136   <.cop 3579   class class class wbr 3982

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1435  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-v 2728  df-br 3983

This theorem is referenced by: (None)