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Theorem eldisjs 38704
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 38690 . 2 Disjs = {𝑟 ∈ Rels ∣ ≀ 𝑟 ∈ CnvRefRels }
2 cnveq 5887 . . . 4 (𝑟 = 𝑅𝑟 = 𝑅)
32cosseqd 38410 . . 3 (𝑟 = 𝑅 → ≀ 𝑟 = ≀ 𝑅)
43eleq1d 2824 . 2 (𝑟 = 𝑅 → ( ≀ 𝑟 ∈ CnvRefRels ↔ ≀ 𝑅 ∈ CnvRefRels ))
51, 4rabeqel 38236 1 (𝑅 ∈ Disjs ↔ ( ≀ 𝑅 ∈ CnvRefRels ∧ 𝑅 ∈ Rels ))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1537  wcel 2106  ccnv 5688  ccoss 38162   Rels crels 38164   CnvRefRels ccnvrefrels 38170   Disjs cdisjs 38195
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-in 3970  df-ss 3980  df-br 5149  df-opab 5211  df-cnv 5697  df-coss 38393  df-disjss 38685  df-disjs 38686
This theorem is referenced by:  eldisjs2  38705  eldisjsdisj  38709
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