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Mirrors > Home > MPE Home > Th. List > Mathboxes > pet0 | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
pet0 | ⊢ (∅ Part 𝐴 ↔ ≀ ∅ ErALTV 𝐴) |
Step | Hyp | Ref | 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)) | |
4 | 1, 2, 3 | 3bitr4i 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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