| 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 37050 uses characterizations of their elements.
Both proofs
then use transitivity of a congruence relation (equality for pw0 4812
and
biconditional for bj-pw0ALT 37050) to translate the property ss0b 4401
into the
wanted result. To translate a biconditional into a class equality, pw0 4812
uses abbii 2809 (which yields an equality of class
abstractions), while
bj-pw0ALT 37050 uses eqriv 2734 (which requires a biconditional of membership
of
a given setvar variable). Note that abbii 2809, through its closed form
abbi 2807, is proved from eqrdv 2735, which is the deduction form of eqriv 2734.
In the other direction, velpw 4605 and velsn 4642 are proved from the
definitions of powerclass and singleton using elabg 3676, which is a version
of abbii 2809 suited for membership characterizations.
(Contributed by BJ,
14-Apr-2024.) (Proof modification is discouraged.)
(New usage is discouraged.) |
