Theorem eldisjsim2 39612

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 4154	. 2 ⊢ (𝑅 ∈ ( Disjss ∩ Rels ) → 𝑅 ∈ Rels )
2		df-disjs 39466	. 2 ⊢ Disjs = ( Disjss ∩ Rels )
3	1, 2	eleq2s 2880	1 ⊢ (𝑅 ∈ Disjs → 𝑅 ∈ Rels )

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2142   ∩ cin 3903   Rels crels 38862   Disjss cdisjss 38894   Disjs cdisjs 38895

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 401  df-tru 1572  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-v 3456  df-in 3911  df-disjs 39466

This theorem is used by:  disjsssrels  39613  eldisjs6  39617