Theorem eldisjs 38714

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 38700	. 2 ⊢ Disjs = {𝑟 ∈ Rels ∣ ≀ ◡𝑟 ∈ CnvRefRels }
2		cnveq 5837	. . . 4 ⊢ (𝑟 = 𝑅 → ◡𝑟 = ◡𝑅)
3	2	cosseqd 38419	. . 3 ⊢ (𝑟 = 𝑅 → ≀ ◡𝑟 = ≀ ◡𝑅)
4	3	eleq1d 2813	. 2 ⊢ (𝑟 = 𝑅 → ( ≀ ◡𝑟 ∈ CnvRefRels ↔ ≀ ◡𝑅 ∈ CnvRefRels ))
5	1, 4	rabeqel 38243	1 ⊢ (𝑅 ∈ Disjs ↔ ( ≀ ◡𝑅 ∈ CnvRefRels ∧ 𝑅 ∈ Rels ))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ^◡ccnv 5637   ≀ ccoss 38169   Rels crels 38171   CnvRefRels ccnvrefrels 38177   Disjs cdisjs 38202

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-rab 3406  df-v 3449  df-in 3921  df-ss 3931  df-br 5108  df-opab 5170  df-cnv 5646  df-coss 38402  df-disjss 38695  df-disjs 38696

This theorem is referenced by:  eldisjs2  38715  eldisjsdisj  38719