Theorem dfdisjs 39483

Description: Alternate definition of the class of disjoints. (Contributed by Peter Mazsa, 18-Jul-2021.)

Assertion

Ref	Expression
dfdisjs	⊢ Disjs = {𝑟 ∈ Rels ∣ ≀ ◡𝑟 ∈ CnvRefRels }

Proof of Theorem dfdisjs

Step	Hyp	Ref	Expression
1		df-disjs 39479	. 2 ⊢ Disjs = ( Disjss ∩ Rels )
2		df-disjss 39478	. 2 ⊢ Disjss = {𝑟 ∣ ≀ ◡𝑟 ∈ CnvRefRels }
3	1, 2	abeqin 38944	1 ⊢ Disjs = {𝑟 ∈ Rels ∣ ≀ ◡𝑟 ∈ CnvRefRels }

This proof depends on syntax axioms:   = wceq 1570   ∈ wcel 2146  {crab 3419  ^◡ccnv 5665   ≀ ccoss 38873   Rels crels 38875   CnvRefRels ccnvrefrels 38881   Disjss cdisjss 38907   Disjs cdisjs 38908

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 2148  ax-9 2156  ax-ext 2738

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-rab 3420  df-v 3460  df-in 3915  df-disjss 39478  df-disjs 39479

This theorem is used by:  dfdisjs2  39484  eldisjs  39509