| Description: Alternate proof of pw0 4762.
The proofs have a similar structure: pw0 4762
uses the definitions of powerclass and singleton as class abstractions,
whereas bj-pw0ALT 37062 uses characterizations of their elements.
Both proofs
then use transitivity of a congruence relation (equality for pw0 4762
and
biconditional for bj-pw0ALT 37062) to translate the property ss0b 4349
into the
wanted result. To translate a biconditional into a class equality, pw0 4762
uses abbii 2797 (which yields an equality of class
abstractions), while
bj-pw0ALT 37062 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 4553 and velsn 4590 are proved from the
definitions of powerclass and singleton using elabg 3630, 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.) |
