| Description: Alternate proof of pw0 4772.
The proofs have a similar structure: pw0 4772
uses the definitions of powerclass and singleton as class abstractions,
whereas bj-pw0ALT 37010 uses characterizations of their elements.
Both proofs
then use transitivity of a congruence relation (equality for pw0 4772
and
biconditional for bj-pw0ALT 37010) to translate the property ss0b 4360
into the
wanted result. To translate a biconditional into a class equality, pw0 4772
uses abbii 2796 (which yields an equality of class
abstractions), while
bj-pw0ALT 37010 uses eqriv 2726 (which requires a biconditional of membership
of
a given setvar variable). Note that abbii 2796, through its closed form
abbi 2794, is proved from eqrdv 2727, which is the deduction form of eqriv 2726.
In the other direction, velpw 4564 and velsn 4601 are proved from the
definitions of powerclass and singleton using elabg 3640, which is a version
of abbii 2796 suited for membership characterizations.
(Contributed by BJ,
14-Apr-2024.) (Proof modification is discouraged.)
(New usage is discouraged.) |
