Theorem eldisjsim2 39868

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 4099	. 2 ⊢ (𝑅 ∈ ( Disjss ∩ Rels ) → 𝑅 ∈ Rels )
2		df-disjs 39722	. 2 ⊢ Disjs = ( Disjss ∩ Rels )
3	1, 2	eleq2s 2878	1 ⊢ (𝑅 ∈ Disjs → 𝑅 ∈ Rels )

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2144   ∩ cin 4089   Rels crels 39118   Disjss cdisjss 39150   Disjs cdisjs 39151

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1572  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-v 3452  df-in 4091  df-disjs 39722

This theorem is used by:  disjsssrels  39869  eldisjs6  39873