Theorem brm 4039

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 3990	. 2 ⊢ (𝐴𝑅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑅)
2		elex2 2746	. 2 ⊢ (⟨𝐴, 𝐵⟩ ∈ 𝑅 → ∃𝑥 𝑥 ∈ 𝑅)
3	1, 2	sylbi 120	1 ⊢ (𝐴𝑅𝐵 → ∃𝑥 𝑥 ∈ 𝑅)

Syntax hints:   → wi 4  ∃wex 1485   ∈ wcel 2141  ⟨cop 3586   class class class wbr 3989

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 1440  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-v 2732  df-br 3990

This theorem is referenced by: (None)