Theorem eldisjsim2 39670

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 4151	. 2 ⊢ (𝑅 ∈ ( Disjss ∩ Rels ) → 𝑅 ∈ Rels )
2		df-disjs 39524	. 2 ⊢ Disjs = ( Disjss ∩ Rels )
3	1, 2	eleq2s 2880	1 ⊢ (𝑅 ∈ Disjs → 𝑅 ∈ Rels )

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2145   ∩ cin 3901   Rels crels 38920   Disjss cdisjss 38952   Disjs cdisjs 38953

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-v 3455  df-in 3909  df-disjs 39524

This theorem is used by:  disjsssrels  39671  eldisjs6  39675