| Description: Alternate proof of pw0 4773.
The proofs have a similar structure: pw0 4773
uses the definitions of powerclass and singleton as class abstractions,
whereas bj-pw0ALT 37546 uses characterizations of their elements.
Both proofs
then use transitivity of a congruence relation (equality for pw0 4773
and
biconditional for bj-pw0ALT 37546) to translate the property ss0b 4358
into the
wanted result. To translate a biconditional into a class equality, pw0 4773
uses abbii 2832 (which yields an equality of class
abstractions), while
bj-pw0ALT 37546 uses eqriv 2762 (which requires a biconditional of membership
of
a given setvar variable). Note that abbii 2832, through its closed form
abbi 2830, is proved from eqrdv 2763, which is the deduction form of eqriv 2762.
In the other direction, velpw 4563 and velsn 4601 are proved from the
definitions of powerclass and singleton using elabg 3638, which is a version
of abbii 2832 suited for membership characterizations.
(Contributed by BJ,
14-Apr-2024.) (Proof modification is discouraged.)
(New usage is discouraged.) |
