Theorem eldisjs 36833

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 36819	. 2 ⊢ Disjs = {𝑟 ∈ Rels ∣ ≀ ◡𝑟 ∈ CnvRefRels }
2		cnveq 5782	. . . 4 ⊢ (𝑟 = 𝑅 → ◡𝑟 = ◡𝑅)
3	2	cosseqd 36551	. . 3 ⊢ (𝑟 = 𝑅 → ≀ ◡𝑟 = ≀ ◡𝑅)
4	3	eleq1d 2823	. 2 ⊢ (𝑟 = 𝑅 → ( ≀ ◡𝑟 ∈ CnvRefRels ↔ ≀ ◡𝑅 ∈ CnvRefRels ))
5	1, 4	rabeqel 36394	1 ⊢ (𝑅 ∈ Disjs ↔ ( ≀ ◡𝑅 ∈ CnvRefRels ∧ 𝑅 ∈ Rels ))

Syntax hints:   ↔ wb 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106  ^◡ccnv 5588   ≀ ccoss 36333   Rels crels 36335   CnvRefRels ccnvrefrels 36341   Disjs cdisjs 36366

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-rab 3073  df-v 3434  df-in 3894  df-ss 3904  df-br 5075  df-opab 5137  df-cnv 5597  df-coss 36537  df-disjss 36814  df-disjs 36815

This theorem is referenced by:  eldisjs2  36834  eldisjsdisj  36838