Theorem brne0 5157

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 5108	. 2 ⊢ (𝐴𝑅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑅)
2		ne0i 4304	. 2 ⊢ (⟨𝐴, 𝐵⟩ ∈ 𝑅 → 𝑅 ≠ ∅)
3	1, 2	sylbi 217	1 ⊢ (𝐴𝑅𝐵 → 𝑅 ≠ ∅)

Syntax hints:   → wi 4   ∈ wcel 2109   ≠ wne 2925  ∅c0 4296  ⟨cop 4595   class class class wbr 5107

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-dif 3917  df-nul 4297  df-br 5108

This theorem is referenced by:  epn0  5543  brfvopabrbr  6965  bropfvvvvlem  8070  brfvimex  44015  brovmptimex  44016  clsneibex  44091  neicvgbex  44101