Theorem brne0 5206

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 5157	. 2 ⊢ (𝐴𝑅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑅)
2		ne0i 4347	. 2 ⊢ (⟨𝐴, 𝐵⟩ ∈ 𝑅 → 𝑅 ≠ ∅)
3	1, 2	sylbi 216	1 ⊢ (𝐴𝑅𝐵 → 𝑅 ≠ ∅)

Syntax hints:   → wi 4   ∈ wcel 2100   ≠ wne 2933  ∅c0 4335  ⟨cop 4642   class class class wbr 5156

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2102  ax-9 2110  ax-ext 2700

This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2062  df-clab 2707  df-cleq 2721  df-clel 2806  df-ne 2934  df-dif 3962  df-nul 4336  df-br 5157

This theorem is referenced by:  epn0  5595  brfvopabrbr  7011  bropfvvvvlem  8113  brfvimex  43764  brovmptimex  43765  clsneibex  43840  neicvgbex  43850