| Description: Alternate proof of pw0 4777.
The proofs have a similar structure: pw0 4777
uses the definitions of powerclass and singleton as class abstractions,
whereas bj-pw0ALT 37651 uses characterizations of their elements.
Both proofs
then use transitivity of a congruence relation (equality for pw0 4777
and
biconditional for bj-pw0ALT 37651) to translate the property ss0b 4357
into the
wanted result. To translate a biconditional into a class equality, pw0 4777
uses abbii 2828 (which yields an equality of class
abstractions), while
bj-pw0ALT 37651 uses eqriv 2758 (which requires a biconditional of membership
of
a given setvar variable). Note that abbii 2828, through its closed form
abbi 2826, is proved from eqrdv 2759, which is the deduction form of eqriv 2758.
In the other direction, velpw 4566 and velsn 4604 are proved from the
definitions of powerclass and singleton using elabg 3634, which is a version
of abbii 2828 suited for membership characterizations.
(Contributed by BJ,
14-Apr-2024.) (Proof modification is discouraged.)
(New usage is discouraged.) |
