| Description: Alternate proof of pw0 4812.
The proofs have a similar structure: pw0 4812
uses the definitions of powerclass and singleton as class abstractions,
whereas bj-pw0ALT 36523 uses characterizations of their elements.
Both proofs
then use transitivity of a congruence relation (equality for pw0 4812
and
biconditional for bj-pw0ALT 36523) to translate the property ss0b 4394
into the
wanted result. To translate a biconditional into a class equality, pw0 4812
uses abbii 2798 (which yields an equality of class
abstractions), while
bj-pw0ALT 36523 uses eqriv 2725 (which requires a biconditional of membership
of
a given setvar variable). Note that abbii 2798, through its closed form
abbi 2796, is proved from eqrdv 2726, which is the deduction form of eqriv 2725.
In the other direction, velpw 4604 and velsn 4641 are proved from the
definitions of powerclass and singleton using elabg 3664, which is a version
of abbii 2798 suited for membership characterizations.
(Contributed by BJ,
14-Apr-2024.) (Proof modification is discouraged.)
(New usage is discouraged.) |
