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Theorem eldisjs5 36876
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 36873 . 2 (𝑅 ∈ Disjs ↔ ( ≀ 𝑅 ⊆ I ∧ 𝑅 ∈ Rels ))
2 cosscnvssid5 36635 . . 3 (( ≀ 𝑅 ⊆ I ∧ Rel 𝑅) ↔ (∀𝑢 ∈ dom 𝑅𝑣 ∈ dom 𝑅(𝑢 = 𝑣 ∨ ([𝑢]𝑅 ∩ [𝑣]𝑅) = ∅) ∧ Rel 𝑅))
3 elrelsrel 36644 . . . . 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 347 . . 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 259 . 2 (𝑅𝑉 → (( ≀ 𝑅 ⊆ I ∧ 𝑅 ∈ Rels ) ↔ (∀𝑢 ∈ dom 𝑅𝑣 ∈ dom 𝑅(𝑢 = 𝑣 ∨ ([𝑢]𝑅 ∩ [𝑣]𝑅) = ∅) ∧ 𝑅 ∈ Rels )))
81, 7bitrid 284 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 205  wa 397  wo 845   = wceq 1539  wcel 2104  wral 3062  cin 3891  wss 3892  c0 4262   I cid 5495  ccnv 5595  dom cdm 5596  Rel wrel 5601  [cec 8523  ccoss 36374   Rels crels 36376   Disjs cdisjs 36407
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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2707  ax-sep 5232  ax-nul 5239  ax-pow 5297  ax-pr 5361  ax-un 7616
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ral 3063  df-rex 3072  df-rmo 3285  df-rab 3287  df-v 3439  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4566  df-pr 4568  df-op 4572  df-uni 4845  df-br 5082  df-opab 5144  df-id 5496  df-xp 5602  df-rel 5603  df-cnv 5604  df-co 5605  df-dm 5606  df-rn 5607  df-res 5608  df-ima 5609  df-ec 8527  df-coss 36576  df-rels 36642  df-ssr 36655  df-cnvrefs 36680  df-cnvrefrels 36681  df-disjss 36853  df-disjs 36854
This theorem is referenced by: (None)
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