Theorem dfdisjs 39818

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 39814	. 2 ⊢ Disjs = ( Disjss ∩ Rels )
2		df-disjss 39813	. 2 ⊢ Disjss = {𝑟 ∣ ≀ ◡𝑟 ∈ CnvRefRels }
3	1, 2	abeqin 39279	1 ⊢ Disjs = {𝑟 ∈ Rels ∣ ≀ ◡𝑟 ∈ CnvRefRels }

This proof depends on syntax axioms:   = wceq 1569   ∈ wcel 2144  {crab 3412  ^◡ccnv 5649   ≀ ccoss 39208   Rels crels 39210   CnvRefRels ccnvrefrels 39216   Disjss cdisjss 39242   Disjs cdisjs 39243

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1572  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-in 4091  df-disjss 39813  df-disjs 39814

This theorem is used by:  dfdisjs2  39819  eldisjs  39844