Theorem brne0 5158

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 5107	. 2 ⊢ (𝐴𝑅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑅)
2		ne0i 4290	. 2 ⊢ (⟨𝐴, 𝐵⟩ ∈ 𝑅 → 𝑅 ≠ ∅)
3	1, 2	sylbi 220	1 ⊢ (𝐴𝑅𝐵 → 𝑅 ≠ ∅)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2146   ≠ wne 2959  ∅c0 4281  ⟨cop 4838   class class class wbr 5106

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2101  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-dif 4050  df-nul 4282  df-br 5107

This theorem is used by:  epn0  5560  brfvopabrbr  7001  bropfvvvvlem  8113  brfvimex  45169  brovmptimex  45170  clsneibex  45245  neicvgbex  45255