Theorem eldisjs 39575

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 39549	. 2 ⊢ Disjs = {𝑟 ∈ Rels ∣ ≀ ◡𝑟 ∈ CnvRefRels }
2		cnveq 5857	. . . 4 ⊢ (𝑟 = 𝑅 → ◡𝑟 = ◡𝑅)
3	2	cosseqd 39274	. . 3 ⊢ (𝑟 = 𝑅 → ≀ ◡𝑟 = ≀ ◡𝑅)
4	3	eleq1d 2847	. 2 ⊢ (𝑟 = 𝑅 → ( ≀ ◡𝑟 ∈ CnvRefRels ↔ ≀ ◡𝑅 ∈ CnvRefRels ))
5	1, 4	rabeqel 39013	1 ⊢ (𝑅 ∈ Disjs ↔ ( ≀ ◡𝑅 ∈ CnvRefRels ∧ 𝑅 ∈ Rels ))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ^◡ccnv 5658   ≀ ccoss 38939   Rels crels 38941   CnvRefRels ccnvrefrels 38947   Disjs cdisjs 38974

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-rab 3415  df-v 3455  df-in 3909  df-ss 3919  df-br 5108  df-opab 5172  df-cnv 5667  df-coss 39257  df-disjss 39544  df-disjs 39545

This theorem is used by:  eldisjs2  39576  eldisjsdisj  39580