| Description: Alternate proof of pw0 4788.
The proofs have a similar structure: pw0 4788
uses the definitions of powerclass and singleton as class abstractions,
whereas bj-pw0ALT 37067 uses characterizations of their elements.
Both proofs
then use transitivity of a congruence relation (equality for pw0 4788
and
biconditional for bj-pw0ALT 37067) to translate the property ss0b 4376
into the
wanted result. To translate a biconditional into a class equality, pw0 4788
uses abbii 2802 (which yields an equality of class
abstractions), while
bj-pw0ALT 37067 uses eqriv 2732 (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 2733, which is the deduction form of eqriv 2732.
In the other direction, velpw 4580 and velsn 4617 are proved from the
definitions of powerclass and singleton using elabg 3655, 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.) |
