| Description: Alternate proof of pw0 4817.
The proofs have a similar structure: pw0 4817
uses the definitions of powerclass and singleton as class abstractions,
whereas bj-pw0ALT 37032 uses characterizations of their elements.
Both proofs
then use transitivity of a congruence relation (equality for pw0 4817
and
biconditional for bj-pw0ALT 37032) to translate the property ss0b 4407
into the
wanted result. To translate a biconditional into a class equality, pw0 4817
uses abbii 2807 (which yields an equality of class
abstractions), while
bj-pw0ALT 37032 uses eqriv 2732 (which requires a biconditional of membership
of
a given setvar variable). Note that abbii 2807, through its closed form
abbi 2805, is proved from eqrdv 2733, which is the deduction form of eqriv 2732.
In the other direction, velpw 4610 and velsn 4647 are proved from the
definitions of powerclass and singleton using elabg 3677, which is a version
of abbii 2807 suited for membership characterizations.
(Contributed by BJ,
14-Apr-2024.) (Proof modification is discouraged.)
(New usage is discouraged.) |
