| Description: Alternate proof of pw0 4816.
The proofs have a similar structure: pw0 4816
uses the definitions of powerclass and singleton as class abstractions,
whereas bj-pw0ALT 35930 uses characterizations of their elements.
Both proofs
then use transitivity of a congruence relation (equality for pw0 4816
and
biconditional for bj-pw0ALT 35930) to translate the property ss0b 4398
into the
wanted result. To translate a biconditional into a class equality, pw0 4816
uses abbii 2803 (which yields an equality of class
abstractions), while
bj-pw0ALT 35930 uses eqriv 2730 (which requires a biconditional of membership
of
a given setvar variable). Note that abbii 2803, through its closed form
abbi 2801, is proved from eqrdv 2731, which is the deduction form of eqriv 2730.
In the other direction, velpw 4608 and velsn 4645 are proved from the
definitions of powerclass and singleton using elabg 3667, which is a version
of abbii 2803 suited for membership characterizations.
(Contributed by BJ,
14-Apr-2024.) (Proof modification is discouraged.)
(New usage is discouraged.) |
