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Theorem pet0 38149
Description: Class 𝐴 is a partition by the null class if and only if the cosets by the null class are in equivalence relation on it. (Contributed by Peter Mazsa, 31-Dec-2021.)
Assertion
Ref Expression
pet0 (∅ Part 𝐴 ↔ ≀ ∅ ErALTV 𝐴)

Proof of Theorem pet0
StepHypRef Expression
1 pet02 38148 . 2 (( Disj ∅ ∧ (dom ∅ / ∅) = 𝐴) ↔ ( EqvRel ≀ ∅ ∧ (dom ≀ ∅ / ≀ ∅) = 𝐴))
2 dfpart2 38103 . 2 (∅ Part 𝐴 ↔ ( Disj ∅ ∧ (dom ∅ / ∅) = 𝐴))
3 dferALTV2 38002 . 2 ( ≀ ∅ ErALTV 𝐴 ↔ ( EqvRel ≀ ∅ ∧ (dom ≀ ∅ / ≀ ∅) = 𝐴))
41, 2, 33bitr4i 303 1 (∅ Part 𝐴 ↔ ≀ ∅ ErALTV 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 395   = wceq 1540  c0 4322  dom cdm 5676   / cqs 8708  ccoss 37507   EqvRel weqvrel 37524   ErALTV werALTV 37533   Disj wdisjALTV 37541   Part wpart 37546
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ral 3061  df-rex 3070  df-rmo 3375  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-ec 8711  df-qs 8715  df-coss 37745  df-refrel 37846  df-cnvrefrel 37861  df-symrel 37878  df-trrel 37908  df-eqvrel 37919  df-dmqs 37973  df-erALTV 37998  df-disjALTV 38039  df-part 38100
This theorem is referenced by: (None)
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