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Theorem dfdisjs5 38693
Description: Alternate definition of the class of disjoints. (Contributed by Peter Mazsa, 5-Sep-2021.)
Assertion
Ref Expression
dfdisjs5 Disjs = {𝑟 ∈ Rels ∣ ∀𝑢 ∈ dom 𝑟𝑣 ∈ dom 𝑟(𝑢 = 𝑣 ∨ ([𝑢]𝑟 ∩ [𝑣]𝑟) = ∅)}
Distinct variable group:   𝑢,𝑟,𝑣

Proof of Theorem dfdisjs5
StepHypRef Expression
1 dfdisjs2 38690 . 2 Disjs = {𝑟 ∈ Rels ∣ ≀ 𝑟 ⊆ I }
2 cosscnvssid5 38459 . . 3 (( ≀ 𝑟 ⊆ I ∧ Rel 𝑟) ↔ (∀𝑢 ∈ dom 𝑟𝑣 ∈ dom 𝑟(𝑢 = 𝑣 ∨ ([𝑢]𝑟 ∩ [𝑣]𝑟) = ∅) ∧ Rel 𝑟))
3 elrelsrelim 38469 . . . . 5 (𝑟 ∈ Rels → Rel 𝑟)
43biantrud 531 . . . 4 (𝑟 ∈ Rels → ( ≀ 𝑟 ⊆ I ↔ ( ≀ 𝑟 ⊆ I ∧ Rel 𝑟)))
53biantrud 531 . . . 4 (𝑟 ∈ Rels → (∀𝑢 ∈ dom 𝑟𝑣 ∈ dom 𝑟(𝑢 = 𝑣 ∨ ([𝑢]𝑟 ∩ [𝑣]𝑟) = ∅) ↔ (∀𝑢 ∈ dom 𝑟𝑣 ∈ dom 𝑟(𝑢 = 𝑣 ∨ ([𝑢]𝑟 ∩ [𝑣]𝑟) = ∅) ∧ Rel 𝑟)))
64, 5bibi12d 345 . . 3 (𝑟 ∈ Rels → (( ≀ 𝑟 ⊆ I ↔ ∀𝑢 ∈ dom 𝑟𝑣 ∈ dom 𝑟(𝑢 = 𝑣 ∨ ([𝑢]𝑟 ∩ [𝑣]𝑟) = ∅)) ↔ (( ≀ 𝑟 ⊆ I ∧ Rel 𝑟) ↔ (∀𝑢 ∈ dom 𝑟𝑣 ∈ dom 𝑟(𝑢 = 𝑣 ∨ ([𝑢]𝑟 ∩ [𝑣]𝑟) = ∅) ∧ Rel 𝑟))))
72, 6mpbiri 258 . 2 (𝑟 ∈ Rels → ( ≀ 𝑟 ⊆ I ↔ ∀𝑢 ∈ dom 𝑟𝑣 ∈ dom 𝑟(𝑢 = 𝑣 ∨ ([𝑢]𝑟 ∩ [𝑣]𝑟) = ∅)))
81, 7rabimbieq 38232 1 Disjs = {𝑟 ∈ Rels ∣ ∀𝑢 ∈ dom 𝑟𝑣 ∈ dom 𝑟(𝑢 = 𝑣 ∨ ([𝑢]𝑟 ∩ [𝑣]𝑟) = ∅)}
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395  wo 847   = wceq 1536  wcel 2105  wral 3058  {crab 3432  cin 3961  wss 3962  c0 4338   I cid 5581  ccnv 5687  dom cdm 5688  Rel wrel 5693  [cec 8741  ccoss 38161   Rels crels 38163   Disjs cdisjs 38194
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ral 3059  df-rex 3068  df-rmo 3377  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-ec 8745  df-coss 38392  df-rels 38466  df-ssr 38479  df-cnvrefs 38506  df-cnvrefrels 38507  df-disjss 38684  df-disjs 38685
This theorem is referenced by: (None)
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