Theorem dfdisjs 38685

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 38681	. 2 ⊢ Disjs = ( Disjss ∩ Rels )
2		df-disjss 38680	. 2 ⊢ Disjss = {𝑟 ∣ ≀ ◡𝑟 ∈ CnvRefRels }
3	1, 2	abeqin 38226	1 ⊢ Disjs = {𝑟 ∈ Rels ∣ ≀ ◡𝑟 ∈ CnvRefRels }

Syntax hints:   = wceq 1540   ∈ wcel 2109  {crab 3396  ^◡ccnv 5622   ≀ ccoss 38154   Rels crels 38156   CnvRefRels ccnvrefrels 38162   Disjss cdisjss 38186   Disjs cdisjs 38187

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-rab 3397  df-v 3440  df-in 3912  df-disjss 38680  df-disjs 38681

This theorem is referenced by:  dfdisjs2  38686  eldisjs  38699