Theorem dfdisjs 39167

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 39163	. 2 ⊢ Disjs = ( Disjss ∩ Rels )
2		df-disjss 39162	. 2 ⊢ Disjss = {𝑟 ∣ ≀ ◡𝑟 ∈ CnvRefRels }
3	1, 2	abeqin 38628	1 ⊢ Disjs = {𝑟 ∈ Rels ∣ ≀ ◡𝑟 ∈ CnvRefRels }

Syntax hints:   = wceq 1547   ∈ wcel 2119  {crab 3392  ^◡ccnv 5624   ≀ ccoss 38557   Rels crels 38559   CnvRefRels ccnvrefrels 38565   Disjss cdisjss 38591   Disjs cdisjs 38592

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2712

This theorem depends on definitions:  df-bi 208  df-an 397  df-tru 1550  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-rab 3393  df-v 3434  df-in 3897  df-disjss 39162  df-disjs 39163

This theorem is referenced by:  dfdisjs2  39168  eldisjs  39193