Theorem brne0 5189

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 5140	. 2 ⊢ (𝐴𝑅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑅)
2		ne0i 4327	. 2 ⊢ (⟨𝐴, 𝐵⟩ ∈ 𝑅 → 𝑅 ≠ ∅)
3	1, 2	sylbi 216	1 ⊢ (𝐴𝑅𝐵 → 𝑅 ≠ ∅)

Syntax hints:   → wi 4   ∈ wcel 2098   ≠ wne 2932  ∅c0 4315  ⟨cop 4627   class class class wbr 5139

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ne 2933  df-dif 3944  df-nul 4316  df-br 5140

This theorem is referenced by:  epn0  5576  brfvopabrbr  6986  bropfvvvvlem  8072  brfvimex  43326  brovmptimex  43327  clsneibex  43402  neicvgbex  43412