Theorem brne0 5163

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 5112	. 2 ⊢ (𝐴𝑅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑅)
2		ne0i 4294	. 2 ⊢ (⟨𝐴, 𝐵⟩ ∈ 𝑅 → 𝑅 ≠ ∅)
3	1, 2	sylbi 220	1 ⊢ (𝐴𝑅𝐵 → 𝑅 ≠ ∅)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2146   ≠ wne 2960  ∅c0 4286  ⟨cop 4597   class class class wbr 5111

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-dif 3909  df-nul 4287  df-br 5112

This theorem is used by:  epn0  5568  brfvopabrbr  6990  bropfvvvvlem  8092  brfvimex  44812  brovmptimex  44813  clsneibex  44888  neicvgbex  44898