| Description: Alternate proof of pw0 4756.
The proofs have a similar structure: pw0 4756
uses the definitions of powerclass and singleton as class abstractions,
whereas bj-pw0ALT 37354 uses characterizations of their elements.
Both proofs
then use transitivity of a congruence relation (equality for pw0 4756
and
biconditional for bj-pw0ALT 37354) to translate the property ss0b 4342
into the
wanted result. To translate a biconditional into a class equality, pw0 4756
uses abbii 2804 (which yields an equality of class
abstractions), while
bj-pw0ALT 37354 uses eqriv 2734 (which requires a biconditional of membership
of
a given setvar variable). Note that abbii 2804, through its closed form
abbi 2802, is proved from eqrdv 2735, which is the deduction form of eqriv 2734.
In the other direction, velpw 4547 and velsn 4584 are proved from the
definitions of powerclass and singleton using elabg 3620, which is a version
of abbii 2804 suited for membership characterizations.
(Contributed by BJ,
14-Apr-2024.) (Proof modification is discouraged.)
(New usage is discouraged.) |
