Theorem eldisjsim2 39439

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 4156	. 2 ⊢ (𝑅 ∈ ( Disjss ∩ Rels ) → 𝑅 ∈ Rels )
2		df-disjs 39293	. 2 ⊢ Disjs = ( Disjss ∩ Rels )
3	1, 2	eleq2s 2882	1 ⊢ (𝑅 ∈ Disjs → 𝑅 ∈ Rels )

Syntax hints:   → wi 4   ∈ wcel 2144   ∩ cin 3905   Rels crels 38689   Disjss cdisjss 38721   Disjs cdisjs 38722

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-ext 2736

This theorem depends on definitions:  df-bi 209  df-an 400  df-tru 1565  df-ex 1802  df-sb 2093  df-clab 2743  df-cleq 2756  df-clel 2839  df-v 3458  df-in 3913  df-disjs 39293

This theorem is referenced by:  disjsssrels  39440  eldisjs6  39444