Theorem brne0 4702

Description: If two sets are in a binary relation, the relation cannot be empty. (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 4654	. 2 |- ( A R B <-> <. A , B >. e. R )
2		ne0i 3921	. 2 |- ( <. A , B >. e. R -> R =/= (/) )
3	1, 2	sylbi 207	1 |- ( A R B -> R =/= (/) )

Syntax hints:   -> wi 4   e. wcel 1990   =/= wne 2794   (/)c0 3915   <.cop 4183   class class class wbr 4653

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-v 3202  df-dif 3577  df-nul 3916  df-br 4654

This theorem is referenced by:  brfvopabrbr  6279  bropfvvvvlem  7256  brfvimex  38324  brovmptimex  38325  clsneibex  38400  neicvgbex  38410