Theorem eldisjsim2 39534

Description: An element of the class of disjoint relations is an element of the class of relations. (Contributed by Peter Mazsa, 11-Feb-2026.)

Assertion

Ref	Expression
eldisjsim2	⊢ (𝑅 ∈ Disjs → 𝑅 ∈ Rels )

Proof of Theorem eldisjsim2

Step	Hyp	Ref	Expression
1		elinel2 4163	. 2 ⊢ (𝑅 ∈ ( Disjss ∩ Rels ) → 𝑅 ∈ Rels )
2		df-disjs 39388	. 2 ⊢ Disjs = ( Disjss ∩ Rels )
3	1, 2	eleq2s 2888	1 ⊢ (𝑅 ∈ Disjs → 𝑅 ∈ Rels )

Syntax hints:   → wi 4   ∈ wcel 2150   ∩ cin 3912   Rels crels 38784   Disjss cdisjss 38816   Disjs cdisjs 38817

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2152  ax-9 2160  ax-ext 2742

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1571  df-ex 1808  df-sb 2099  df-clab 2749  df-cleq 2762  df-clel 2845  df-v 3464  df-in 3920  df-disjs 39388

This theorem is referenced by:  disjsssrels  39535  eldisjs6  39539