Theorem eldisjsim2 39138

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 4155	. 2 ⊢ (𝑅 ∈ ( Disjss ∩ Rels ) → 𝑅 ∈ Rels )
2		df-disjs 38992	. 2 ⊢ Disjs = ( Disjss ∩ Rels )
3	1, 2	eleq2s 2855	1 ⊢ (𝑅 ∈ Disjs → 𝑅 ∈ Rels )

Syntax hints:   → wi 4   ∈ wcel 2114   ∩ cin 3901   Rels crels 38388   Disjss cdisjss 38420   Disjs cdisjs 38421

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-v 3443  df-in 3909  df-disjs 38992

This theorem is referenced by:  disjsssrels  39139  eldisjs6  39143