| Description: Alternate proof of pw0 4779.
The proofs have a similar structure: pw0 4779
uses the definitions of powerclass and singleton as class abstractions,
whereas bj-pw0ALT 37044 uses characterizations of their elements.
Both proofs
then use transitivity of a congruence relation (equality for pw0 4779
and
biconditional for bj-pw0ALT 37044) to translate the property ss0b 4367
into the
wanted result. To translate a biconditional into a class equality, pw0 4779
uses abbii 2797 (which yields an equality of class
abstractions), while
bj-pw0ALT 37044 uses eqriv 2727 (which requires a biconditional of membership
of
a given setvar variable). Note that abbii 2797, through its closed form
abbi 2795, is proved from eqrdv 2728, which is the deduction form of eqriv 2727.
In the other direction, velpw 4571 and velsn 4608 are proved from the
definitions of powerclass and singleton using elabg 3646, which is a version
of abbii 2797 suited for membership characterizations.
(Contributed by BJ,
14-Apr-2024.) (Proof modification is discouraged.)
(New usage is discouraged.) |
