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Theorem eldisjs 38723
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.
StepHypRef Expression
1 dfdisjs 38709 . 2 Disjs = {𝑟 ∈ Rels ∣ ≀ 𝑟 ∈ CnvRefRels }
2 cnveq 5884 . . . 4 (𝑟 = 𝑅𝑟 = 𝑅)
32cosseqd 38429 . . 3 (𝑟 = 𝑅 → ≀ 𝑟 = ≀ 𝑅)
43eleq1d 2826 . 2 (𝑟 = 𝑅 → ( ≀ 𝑟 ∈ CnvRefRels ↔ ≀ 𝑅 ∈ CnvRefRels ))
51, 4rabeqel 38255 1 (𝑅 ∈ Disjs ↔ ( ≀ 𝑅 ∈ CnvRefRels ∧ 𝑅 ∈ Rels ))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1540  wcel 2108  ccnv 5684  ccoss 38182   Rels crels 38184   CnvRefRels ccnvrefrels 38190   Disjs cdisjs 38215
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 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3437  df-v 3482  df-in 3958  df-ss 3968  df-br 5144  df-opab 5206  df-cnv 5693  df-coss 38412  df-disjss 38704  df-disjs 38705
This theorem is referenced by:  eldisjs2  38724  eldisjsdisj  38728
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