Metamath Proof Explorer


Theorem dfmembpart2

Description: Alternate definition of the conventional membership case of partition. Partition A of X , Halmos p. 28: "A partition of X is a disjoint collection A of non-empty subsets of X whose union is X ", or Definition 35, Suppes p. 83., cf. https://oeis.org/A000110 . (Contributed by Peter Mazsa, 14-Aug-2021)

Ref Expression
Assertion dfmembpart2 MembPart A ElDisj A ¬ A

Proof

Step Hyp Ref Expression
1 df-membpart MembPart A E -1 A Part A
2 df-part E -1 A Part A Disj E -1 A E -1 A DomainQs A
3 df-eldisj ElDisj A Disj E -1 A
4 3 bicomi Disj E -1 A ElDisj A
5 cnvepresdmqs E -1 A DomainQs A ¬ A
6 4 5 anbi12i Disj E -1 A E -1 A DomainQs A ElDisj A ¬ A
7 1 2 6 3bitri MembPart A ElDisj A ¬ A