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Theorem eldisjs 37640
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 37626 . 2 Disjs = {𝑟 ∈ Rels ∣ ≀ 𝑟 ∈ CnvRefRels }
2 cnveq 5874 . . . 4 (𝑟 = 𝑅𝑟 = 𝑅)
32cosseqd 37346 . . 3 (𝑟 = 𝑅 → ≀ 𝑟 = ≀ 𝑅)
43eleq1d 2819 . 2 (𝑟 = 𝑅 → ( ≀ 𝑟 ∈ CnvRefRels ↔ ≀ 𝑅 ∈ CnvRefRels ))
51, 4rabeqel 37170 1 (𝑅 ∈ Disjs ↔ ( ≀ 𝑅 ∈ CnvRefRels ∧ 𝑅 ∈ Rels ))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 397   = wceq 1542  wcel 2107  ccnv 5676  ccoss 37091   Rels crels 37093   CnvRefRels ccnvrefrels 37099   Disjs cdisjs 37124
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3434  df-v 3477  df-in 3956  df-ss 3966  df-br 5150  df-opab 5212  df-cnv 5685  df-coss 37329  df-disjss 37621  df-disjs 37622
This theorem is referenced by:  eldisjs2  37641  eldisjsdisj  37645
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