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Theorem pet0 38179
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 38178 . 2 (( Disj ∅ ∧ (dom ∅ / ∅) = 𝐴) ↔ ( EqvRel ≀ ∅ ∧ (dom ≀ ∅ / ≀ ∅) = 𝐴))
2 dfpart2 38133 . 2 (∅ Part 𝐴 ↔ ( Disj ∅ ∧ (dom ∅ / ∅) = 𝐴))
3 dferALTV2 38032 . 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 1533  c0 4315  dom cdm 5667   / cqs 8699  ccoss 37537   EqvRel weqvrel 37554   ErALTV werALTV 37563   Disj wdisjALTV 37571   Part wpart 37576
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pr 5418
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ral 3054  df-rex 3063  df-rmo 3368  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-br 5140  df-opab 5202  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-ec 8702  df-qs 8706  df-coss 37775  df-refrel 37876  df-cnvrefrel 37891  df-symrel 37908  df-trrel 37938  df-eqvrel 37949  df-dmqs 38003  df-erALTV 38028  df-disjALTV 38069  df-part 38130
This theorem is referenced by: (None)
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