Theorem eldisjs 36760

Description: Elementhood in the class of disjoints. (Contributed by Peter Mazsa, 24-Jul-2021.)

Assertion

Ref	Expression
eldisjs	⊢ (𝑅 ∈ Disjs ↔ ( ≀ ◡𝑅 ∈ CnvRefRels ∧ 𝑅 ∈ Rels ))

Proof of Theorem eldisjs

Dummy variable 𝑟 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfdisjs 36746	. 2 ⊢ Disjs = {𝑟 ∈ Rels ∣ ≀ ◡𝑟 ∈ CnvRefRels }
2		cnveq 5771	. . . 4 ⊢ (𝑟 = 𝑅 → ◡𝑟 = ◡𝑅)
3	2	cosseqd 36478	. . 3 ⊢ (𝑟 = 𝑅 → ≀ ◡𝑟 = ≀ ◡𝑅)
4	3	eleq1d 2823	. 2 ⊢ (𝑟 = 𝑅 → ( ≀ ◡𝑟 ∈ CnvRefRels ↔ ≀ ◡𝑅 ∈ CnvRefRels ))
5	1, 4	rabeqel 36321	1 ⊢ (𝑅 ∈ Disjs ↔ ( ≀ ◡𝑅 ∈ CnvRefRels ∧ 𝑅 ∈ Rels ))

Syntax hints:   ↔ wb 205   ∧ wa 395   = wceq 1539   ∈ wcel 2108  ^◡ccnv 5579   ≀ ccoss 36260   Rels crels 36262   CnvRefRels ccnvrefrels 36268   Disjs cdisjs 36293

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1542  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-rab 3072  df-v 3424  df-in 3890  df-ss 3900  df-br 5071  df-opab 5133  df-cnv 5588  df-coss 36464  df-disjss 36741  df-disjs 36742

This theorem is referenced by:  eldisjs2  36761  eldisjsdisj  36765