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Theorem pet0 38282
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 38281 . 2 (( Disj ∅ ∧ (dom ∅ / ∅) = 𝐴) ↔ ( EqvRel ≀ ∅ ∧ (dom ≀ ∅ / ≀ ∅) = 𝐴))
2 dfpart2 38236 . 2 (∅ Part 𝐴 ↔ ( Disj ∅ ∧ (dom ∅ / ∅) = 𝐴))
3 dferALTV2 38135 . 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 1534  c0 4319  dom cdm 5673   / cqs 8718  ccoss 37643   EqvRel weqvrel 37660   ErALTV werALTV 37669   Disj wdisjALTV 37677   Part wpart 37682
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5294  ax-nul 5301  ax-pr 5424
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ral 3058  df-rex 3067  df-rmo 3372  df-rab 3429  df-v 3472  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4320  df-if 4526  df-sn 4626  df-pr 4628  df-op 4632  df-br 5144  df-opab 5206  df-id 5571  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  df-ec 8721  df-qs 8725  df-coss 37878  df-refrel 37979  df-cnvrefrel 37994  df-symrel 38011  df-trrel 38041  df-eqvrel 38052  df-dmqs 38106  df-erALTV 38131  df-disjALTV 38172  df-part 38233
This theorem is referenced by: (None)
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