Theorem brne0 4038

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 4039. (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 3990	. 2 |- ( A R B <-> <. A , B >. e. R )
2		ne0i 3421	. 2 |- ( <. A , B >. e. R -> R =/= (/) )
3	1, 2	sylbi 120	1 |- ( A R B -> R =/= (/) )

Syntax hints:   -> wi 4   e. wcel 2141   =/= wne 2340   (/)c0 3414   <.cop 3586   class class class wbr 3989

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-v 2732  df-dif 3123  df-nul 3415  df-br 3990

This theorem is referenced by: (None)