| Description: Alternate proof of pw0 4766.
The proofs have a similar structure: pw0 4766
uses the definitions of powerclass and singleton as class abstractions,
whereas bj-pw0ALT 37193 uses characterizations of their elements.
Both proofs
then use transitivity of a congruence relation (equality for pw0 4766
and
biconditional for bj-pw0ALT 37193) to translate the property ss0b 4351
into the
wanted result. To translate a biconditional into a class equality, pw0 4766
uses abbii 2801 (which yields an equality of class
abstractions), while
bj-pw0ALT 37193 uses eqriv 2731 (which requires a biconditional of membership
of
a given setvar variable). Note that abbii 2801, through its closed form
abbi 2799, is proved from eqrdv 2732, which is the deduction form of eqriv 2731.
In the other direction, velpw 4557 and velsn 4594 are proved from the
definitions of powerclass and singleton using elabg 3629, which is a version
of abbii 2801 suited for membership characterizations.
(Contributed by BJ,
14-Apr-2024.) (Proof modification is discouraged.)
(New usage is discouraged.) |
