Theorem eldisjs 38678

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 38664	. 2 ⊢ Disjs = {𝑟 ∈ Rels ∣ ≀ ◡𝑟 ∈ CnvRefRels }
2		cnveq 5898	. . . 4 ⊢ (𝑟 = 𝑅 → ◡𝑟 = ◡𝑅)
3	2	cosseqd 38384	. . 3 ⊢ (𝑟 = 𝑅 → ≀ ◡𝑟 = ≀ ◡𝑅)
4	3	eleq1d 2829	. 2 ⊢ (𝑟 = 𝑅 → ( ≀ ◡𝑟 ∈ CnvRefRels ↔ ≀ ◡𝑅 ∈ CnvRefRels ))
5	1, 4	rabeqel 38210	1 ⊢ (𝑅 ∈ Disjs ↔ ( ≀ ◡𝑅 ∈ CnvRefRels ∧ 𝑅 ∈ Rels ))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2108  ^◡ccnv 5699   ≀ ccoss 38135   Rels crels 38137   CnvRefRels ccnvrefrels 38143   Disjs cdisjs 38168

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-rab 3444  df-v 3490  df-in 3983  df-ss 3993  df-br 5167  df-opab 5229  df-cnv 5708  df-coss 38367  df-disjss 38659  df-disjs 38660

This theorem is referenced by:  eldisjs2  38679  eldisjsdisj  38683