| Description: Alternate proof of pw0 4815.
The proofs have a similar structure: pw0 4815
uses the definitions of powerclass and singleton as class abstractions,
whereas bj-pw0ALT 36016 uses characterizations of their elements.
Both proofs
then use transitivity of a congruence relation (equality for pw0 4815
and
biconditional for bj-pw0ALT 36016) to translate the property ss0b 4397
into the
wanted result. To translate a biconditional into a class equality, pw0 4815
uses abbii 2802 (which yields an equality of class
abstractions), while
bj-pw0ALT 36016 uses eqriv 2729 (which requires a biconditional of membership
of
a given setvar variable). Note that abbii 2802, through its closed form
abbi 2800, is proved from eqrdv 2730, which is the deduction form of eqriv 2729.
In the other direction, velpw 4607 and velsn 4644 are proved from the
definitions of powerclass and singleton using elabg 3666, which is a version
of abbii 2802 suited for membership characterizations.
(Contributed by BJ,
14-Apr-2024.) (Proof modification is discouraged.)
(New usage is discouraged.) |
