Theorem brne0 5216

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	⊢ (𝐴𝑅𝐵 → 𝑅 ≠ ∅)

Proof of Theorem brne0

Step	Hyp	Ref	Expression
1		df-br 5167	. 2 ⊢ (𝐴𝑅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑅)
2		ne0i 4364	. 2 ⊢ (⟨𝐴, 𝐵⟩ ∈ 𝑅 → 𝑅 ≠ ∅)
3	1, 2	sylbi 217	1 ⊢ (𝐴𝑅𝐵 → 𝑅 ≠ ∅)

Syntax hints:   → wi 4   ∈ wcel 2108   ≠ wne 2946  ∅c0 4352  ⟨cop 4654   class class class wbr 5166

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ne 2947  df-dif 3979  df-nul 4353  df-br 5167

This theorem is referenced by:  epn0  5604  brfvopabrbr  7026  bropfvvvvlem  8132  brfvimex  43988  brovmptimex  43989  clsneibex  44064  neicvgbex  44074