Theorem brm 4079

Description: If two sets are in a binary relation, the relation is inhabited. (Contributed by Jim Kingdon, 31-Dec-2023.)

Assertion

Ref	Expression
brm	⊢ (𝐴𝑅𝐵 → ∃𝑥 𝑥 ∈ 𝑅)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝑅

Proof of Theorem brm

Step	Hyp	Ref	Expression
1		df-br 4030	. 2 ⊢ (𝐴𝑅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑅)
2		elex2 2776	. 2 ⊢ (⟨𝐴, 𝐵⟩ ∈ 𝑅 → ∃𝑥 𝑥 ∈ 𝑅)
3	1, 2	sylbi 121	1 ⊢ (𝐴𝑅𝐵 → ∃𝑥 𝑥 ∈ 𝑅)

Syntax hints:   → wi 4  ∃wex 1503   ∈ wcel 2164  ⟨cop 3621   class class class wbr 4029

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1458  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-v 2762  df-br 4030

This theorem is referenced by:  elfvm  5587