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Theorem eldisjs 35841
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 35827 . 2 Disjs = {𝑟 ∈ Rels ∣ ≀ 𝑟 ∈ CnvRefRels }
2 cnveq 5743 . . . 4 (𝑟 = 𝑅𝑟 = 𝑅)
32cosseqd 35559 . . 3 (𝑟 = 𝑅 → ≀ 𝑟 = ≀ 𝑅)
43eleq1d 2902 . 2 (𝑟 = 𝑅 → ( ≀ 𝑟 ∈ CnvRefRels ↔ ≀ 𝑅 ∈ CnvRefRels ))
51, 4rabeqel 35403 1 (𝑅 ∈ Disjs ↔ ( ≀ 𝑅 ∈ CnvRefRels ∧ 𝑅 ∈ Rels ))
Colors of variables: wff setvar class
Syntax hints:  wb 207  wa 396   = wceq 1530  wcel 2107  ccnv 5553  ccoss 35340   Rels crels 35342   CnvRefRels ccnvrefrels 35348   Disjs cdisjs 35373
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-rab 3152  df-v 3502  df-in 3947  df-ss 3956  df-br 5064  df-opab 5126  df-cnv 5562  df-coss 35545  df-disjss 35822  df-disjs 35823
This theorem is referenced by:  eldisjs2  35842  eldisjsdisj  35846
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