| Description: Alternate proof of pw0 4764.
The proofs have a similar structure: pw0 4764
uses the definitions of powerclass and singleton as class abstractions,
whereas bj-pw0ALT 37482 uses characterizations of their elements.
Both proofs
then use transitivity of a congruence relation (equality for pw0 4764
and
biconditional for bj-pw0ALT 37482) to translate the property ss0b 4349
into the
wanted result. To translate a biconditional into a class equality, pw0 4764
uses abbii 2823 (which yields an equality of class
abstractions), while
bj-pw0ALT 37482 uses eqriv 2753 (which requires a biconditional of membership
of
a given setvar variable). Note that abbii 2823, through its closed form
abbi 2821, is proved from eqrdv 2754, which is the deduction form of eqriv 2753.
In the other direction, velpw 4554 and velsn 4592 are proved from the
definitions of powerclass and singleton using elabg 3630, which is a version
of abbii 2823 suited for membership characterizations.
(Contributed by BJ,
14-Apr-2024.) (Proof modification is discouraged.)
(New usage is discouraged.) |
