Theorem eldisjsim2 39317

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 4134	. 2 ⊢ (𝑅 ∈ ( Disjss ∩ Rels ) → 𝑅 ∈ Rels )
2		df-disjs 39171	. 2 ⊢ Disjs = ( Disjss ∩ Rels )
3	1, 2	eleq2s 2859	1 ⊢ (𝑅 ∈ Disjs → 𝑅 ∈ Rels )

Syntax hints:   → wi 4   ∈ wcel 2121   ∩ cin 3884   Rels crels 38567   Disjss cdisjss 38599   Disjs cdisjs 38600

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-ext 2713

This theorem depends on definitions:  df-bi 209  df-an 398  df-tru 1551  df-ex 1788  df-sb 2075  df-clab 2720  df-cleq 2733  df-clel 2816  df-v 3435  df-in 3892  df-disjs 39171

This theorem is referenced by:  disjsssrels  39318  eldisjs6  39322