| Description: Alternate proof of pw0 4781.
The proofs have a similar structure: pw0 4781
uses the definitions of powerclass and singleton as class abstractions,
whereas bj-pw0ALT 37717 uses characterizations of their elements.
Both proofs
then use transitivity of a congruence relation (equality for pw0 4781
and
biconditional for bj-pw0ALT 37717) to translate the property ss0b 4361
into the
wanted result. To translate a biconditional into a class equality, pw0 4781
uses abbii 2833 (which yields an equality of class
abstractions), while
bj-pw0ALT 37717 uses eqriv 2763 (which requires a biconditional of membership
of
a given setvar variable). Note that abbii 2833, through its closed form
abbi 2831, is proved from eqrdv 2764, which is the deduction form of eqriv 2763.
In the other direction, velpw 4570 and velsn 4608 are proved from the
definitions of powerclass and singleton using elabg 3638, which is a version
of abbii 2833 suited for membership characterizations.
(Contributed by BJ,
14-Apr-2024.) (Proof modification is discouraged.)
(New usage is discouraged.) |
