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Theorem brm 3978
 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 𝑥𝑅)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝑅

Proof of Theorem brm
StepHypRef Expression
1 df-br 3930 . 2 (𝐴𝑅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑅)
2 elex2 2702 . 2 (⟨𝐴, 𝐵⟩ ∈ 𝑅 → ∃𝑥 𝑥𝑅)
31, 2sylbi 120 1 (𝐴𝑅𝐵 → ∃𝑥 𝑥𝑅)
 Colors of variables: wff set class Syntax hints:   → wi 4  ∃wex 1468   ∈ wcel 1480  ⟨cop 3530   class class class wbr 3929 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 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-ext 2121 This theorem depends on definitions:  df-bi 116  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-v 2688  df-br 3930 This theorem is referenced by: (None)
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