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Theorem eldisjs5 38727
Description: Elementhood in the class of disjoints. (Contributed by Peter Mazsa, 5-Sep-2021.)
Assertion
Ref Expression
eldisjs5 (𝑅𝑉 → (𝑅 ∈ Disjs ↔ (∀𝑢 ∈ dom 𝑅𝑣 ∈ dom 𝑅(𝑢 = 𝑣 ∨ ([𝑢]𝑅 ∩ [𝑣]𝑅) = ∅) ∧ 𝑅 ∈ Rels )))
Distinct variable group:   𝑢,𝑅,𝑣
Allowed substitution hints:   𝑉(𝑣,𝑢)

Proof of Theorem eldisjs5
StepHypRef Expression
1 eldisjs2 38724 . 2 (𝑅 ∈ Disjs ↔ ( ≀ 𝑅 ⊆ I ∧ 𝑅 ∈ Rels ))
2 cosscnvssid5 38479 . . 3 (( ≀ 𝑅 ⊆ I ∧ Rel 𝑅) ↔ (∀𝑢 ∈ dom 𝑅𝑣 ∈ dom 𝑅(𝑢 = 𝑣 ∨ ([𝑢]𝑅 ∩ [𝑣]𝑅) = ∅) ∧ Rel 𝑅))
3 elrelsrel 38488 . . . . 5 (𝑅𝑉 → (𝑅 ∈ Rels ↔ Rel 𝑅))
43anbi2d 630 . . . 4 (𝑅𝑉 → (( ≀ 𝑅 ⊆ I ∧ 𝑅 ∈ Rels ) ↔ ( ≀ 𝑅 ⊆ I ∧ Rel 𝑅)))
53anbi2d 630 . . . 4 (𝑅𝑉 → ((∀𝑢 ∈ dom 𝑅𝑣 ∈ dom 𝑅(𝑢 = 𝑣 ∨ ([𝑢]𝑅 ∩ [𝑣]𝑅) = ∅) ∧ 𝑅 ∈ Rels ) ↔ (∀𝑢 ∈ dom 𝑅𝑣 ∈ dom 𝑅(𝑢 = 𝑣 ∨ ([𝑢]𝑅 ∩ [𝑣]𝑅) = ∅) ∧ Rel 𝑅)))
64, 5bibi12d 345 . . 3 (𝑅𝑉 → ((( ≀ 𝑅 ⊆ I ∧ 𝑅 ∈ Rels ) ↔ (∀𝑢 ∈ dom 𝑅𝑣 ∈ dom 𝑅(𝑢 = 𝑣 ∨ ([𝑢]𝑅 ∩ [𝑣]𝑅) = ∅) ∧ 𝑅 ∈ Rels )) ↔ (( ≀ 𝑅 ⊆ I ∧ Rel 𝑅) ↔ (∀𝑢 ∈ dom 𝑅𝑣 ∈ dom 𝑅(𝑢 = 𝑣 ∨ ([𝑢]𝑅 ∩ [𝑣]𝑅) = ∅) ∧ Rel 𝑅))))
72, 6mpbiri 258 . 2 (𝑅𝑉 → (( ≀ 𝑅 ⊆ I ∧ 𝑅 ∈ Rels ) ↔ (∀𝑢 ∈ dom 𝑅𝑣 ∈ dom 𝑅(𝑢 = 𝑣 ∨ ([𝑢]𝑅 ∩ [𝑣]𝑅) = ∅) ∧ 𝑅 ∈ Rels )))
81, 7bitrid 283 1 (𝑅𝑉 → (𝑅 ∈ Disjs ↔ (∀𝑢 ∈ dom 𝑅𝑣 ∈ dom 𝑅(𝑢 = 𝑣 ∨ ([𝑢]𝑅 ∩ [𝑣]𝑅) = ∅) ∧ 𝑅 ∈ Rels )))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wo 848   = wceq 1540  wcel 2108  wral 3061  cin 3950  wss 3951  c0 4333   I cid 5577  ccnv 5684  dom cdm 5685  Rel wrel 5690  [cec 8743  ccoss 38182   Rels crels 38184   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-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rmo 3380  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-ec 8747  df-coss 38412  df-rels 38486  df-ssr 38499  df-cnvrefs 38526  df-cnvrefrels 38527  df-disjss 38704  df-disjs 38705
This theorem is referenced by: (None)
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