Theorem brm 3986

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 3938	. 2 ⊢ (𝐴𝑅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑅)
2		elex2 2705	. 2 ⊢ (⟨𝐴, 𝐵⟩ ∈ 𝑅 → ∃𝑥 𝑥 ∈ 𝑅)
3	1, 2	sylbi 120	1 ⊢ (𝐴𝑅𝐵 → ∃𝑥 𝑥 ∈ 𝑅)

Syntax hints:   → wi 4  ∃wex 1469   ∈ wcel 1481  ⟨cop 3535   class class class wbr 3937

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1424  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-v 2691  df-br 3938

This theorem is referenced by: (None)