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Theorem brm 4079
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
StepHypRef Expression
1 df-br 4030 . 2  |-  ( A R B  <->  <. A ,  B >.  e.  R )
2 elex2 2776 . 2  |-  ( <. A ,  B >.  e.  R  ->  E. x  x  e.  R )
31, 2sylbi 121 1  |-  ( A R B  ->  E. x  x  e.  R )
Colors of variables: wff set class
Syntax hints:    -> wi 4   E.wex 1503    e. wcel 2164   <.cop 3621   class class class wbr 4029
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 1458  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-ext 2175
This theorem depends on definitions:  df-bi 117  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-v 2762  df-br 4030
This theorem is referenced by:  elfvm  5579
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