Theorem brm 4083

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 4034	. 2 ⊢ (𝐴𝑅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑅)
2		elex2 2779	. 2 ⊢ (⟨𝐴, 𝐵⟩ ∈ 𝑅 → ∃𝑥 𝑥 ∈ 𝑅)
3	1, 2	sylbi 121	1 ⊢ (𝐴𝑅𝐵 → ∃𝑥 𝑥 ∈ 𝑅)

Syntax hints:   → wi 4  ∃wex 1506   ∈ wcel 2167  ⟨cop 3625   class class class wbr 4033

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 1461  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-v 2765  df-br 4034

This theorem is referenced by:  elfvm  5591