Theorem brne0 4082

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 4083. (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 4034	. 2 |- ( A R B <-> <. A , B >. e. R )
2		ne0i 3457	. 2 |- ( <. A , B >. e. R -> R =/= (/) )
3	1, 2	sylbi 121	1 |- ( A R B -> R =/= (/) )

Syntax hints:   -> wi 4   e. wcel 2167   =/= wne 2367   (/)c0 3450   <.cop 3625   class class class wbr 4033

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-in1 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-v 2765  df-dif 3159  df-nul 3451  df-br 4034

This theorem is referenced by: (None)