Theorem dfdisjs 39928

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 39924	. 2 ⊢ Disjs = ( Disjss ∩ Rels )
2		df-disjss 39923	. 2 ⊢ Disjss = {𝑟 ∣ ≀ ◡𝑟 ∈ CnvRefRels }
3	1, 2	abeqin 39389	1 ⊢ Disjs = {𝑟 ∈ Rels ∣ ≀ ◡𝑟 ∈ CnvRefRels }

This proof depends on syntax axioms:   = wceq 1570   ∈ wcel 2147  {crab 3416  ^◡ccnv 5655   ≀ ccoss 39318   Rels crels 39320   CnvRefRels ccnvrefrels 39326   Disjss cdisjss 39352   Disjs cdisjs 39353

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-rab 3417  df-v 3456  df-in 4094  df-disjss 39923  df-disjs 39924

This theorem is used by:  dfdisjs2  39929  eldisjs  39954