Theorem brne0 4031

Description: If two sets are in a binary relation, the relation cannot be empty. In fact, the relation is also inhabited, as seen at brm 4032. (Contributed by Alexander van der Vekens, 7-Jul-2018.)

Assertion

Ref	Expression
brne0	|- ( A R B -> R =/= (/) )

Proof of Theorem brne0

Step	Hyp	Ref	Expression
1		df-br 3983	. 2 |- ( A R B <-> <. A , B >. e. R )
2		ne0i 3415	. 2 |- ( <. A , B >. e. R -> R =/= (/) )
3	1, 2	sylbi 120	1 |- ( A R B -> R =/= (/) )

Syntax hints:   -> wi 4   e. wcel 2136   =/= wne 2336   (/)c0 3409   <.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-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-v 2728  df-dif 3118  df-nul 3410  df-br 3983

This theorem is referenced by: (None)