Theorem dfdisjs 39693

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 39689	. 2 ⊢ Disjs = ( Disjss ∩ Rels )
2		df-disjss 39688	. 2 ⊢ Disjss = {𝑟 ∣ ≀ ◡𝑟 ∈ CnvRefRels }
3	1, 2	abeqin 39154	1 ⊢ Disjs = {𝑟 ∈ Rels ∣ ≀ ◡𝑟 ∈ CnvRefRels }

This proof depends on syntax axioms:   = wceq 1570   ∈ wcel 2145  {crab 3413  ^◡ccnv 5650   ≀ ccoss 39083   Rels crels 39085   CnvRefRels ccnvrefrels 39091   Disjss cdisjss 39117   Disjs cdisjs 39118

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 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-rab 3414  df-v 3453  df-in 3906  df-disjss 39688  df-disjs 39689

This theorem is used by:  dfdisjs2  39694  eldisjs  39719