Theorem dfdisjs 39128

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 39124	. 2 ⊢ Disjs = ( Disjss ∩ Rels )
2		df-disjss 39123	. 2 ⊢ Disjss = {𝑟 ∣ ≀ ◡𝑟 ∈ CnvRefRels }
3	1, 2	abeqin 38589	1 ⊢ Disjs = {𝑟 ∈ Rels ∣ ≀ ◡𝑟 ∈ CnvRefRels }

Syntax hints:   = wceq 1542   ∈ wcel 2114  {crab 3390  ^◡ccnv 5623   ≀ ccoss 38518   Rels crels 38520   CnvRefRels ccnvrefrels 38526   Disjss cdisjss 38552   Disjs cdisjs 38553

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-rab 3391  df-v 3432  df-in 3897  df-disjss 39123  df-disjs 39124

This theorem is referenced by:  dfdisjs2  39129  eldisjs  39154