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Mirrors > Home > MPE Home > Th. List > Mathboxes > pet02 | 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 |
---|---|
pet02 | ⊢ (( Disj ∅ ∧ (dom ∅ / ∅) = 𝐴) ↔ ( EqvRel ≀ ∅ ∧ (dom ≀ ∅ / ≀ ∅) = 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | disjALTV0 37493 | . 2 ⊢ Disj ∅ | |
2 | 1 | petlemi 37552 | 1 ⊢ (( Disj ∅ ∧ (dom ∅ / ∅) = 𝐴) ↔ ( EqvRel ≀ ∅ ∧ (dom ≀ ∅ / ≀ ∅) = 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 396 = wceq 1541 ∅c0 4319 dom cdm 5670 / cqs 8687 ≀ ccoss 36912 EqvRel weqvrel 36929 Disj wdisjALTV 36946 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5293 ax-nul 5300 ax-pr 5421 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ral 3062 df-rex 3071 df-rmo 3376 df-rab 3433 df-v 3476 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4320 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-br 5143 df-opab 5205 df-id 5568 df-xp 5676 df-rel 5677 df-cnv 5678 df-co 5679 df-dm 5680 df-rn 5681 df-res 5682 df-ima 5683 df-ec 8690 df-qs 8694 df-coss 37150 df-refrel 37251 df-cnvrefrel 37266 df-symrel 37283 df-trrel 37313 df-eqvrel 37324 df-disjALTV 37444 |
This theorem is referenced by: pet0 37554 |
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