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Theorem eldisjs 37580
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 37566 . 2 Disjs = {𝑟 ∈ Rels ∣ ≀ 𝑟 ∈ CnvRefRels }
2 cnveq 5871 . . . 4 (𝑟 = 𝑅𝑟 = 𝑅)
32cosseqd 37286 . . 3 (𝑟 = 𝑅 → ≀ 𝑟 = ≀ 𝑅)
43eleq1d 2818 . 2 (𝑟 = 𝑅 → ( ≀ 𝑟 ∈ CnvRefRels ↔ ≀ 𝑅 ∈ CnvRefRels ))
51, 4rabeqel 37110 1 (𝑅 ∈ Disjs ↔ ( ≀ 𝑅 ∈ CnvRefRels ∧ 𝑅 ∈ Rels ))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 396   = wceq 1541  wcel 2106  ccnv 5674  ccoss 37031   Rels crels 37033   CnvRefRels ccnvrefrels 37039   Disjs cdisjs 37064
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-rab 3433  df-v 3476  df-in 3954  df-ss 3964  df-br 5148  df-opab 5210  df-cnv 5683  df-coss 37269  df-disjss 37561  df-disjs 37562
This theorem is referenced by:  eldisjs2  37581  eldisjsdisj  37585
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