| Description: Alternate proof of pw0 4745.
The proofs have a similar structure: pw0 4745
uses the definitions of powerclass and singleton as class abstractions,
whereas bj-pw0ALT 35222 uses characterizations of their elements.
Both proofs
then use transitivity of a congruence relation (equality for pw0 4745
and
biconditional for bj-pw0ALT 35222) to translate the property ss0b 4331
into the
wanted result. To translate a biconditional into a class equality, pw0 4745
uses abbii 2808 (which yields an equality of class
abstractions), while
bj-pw0ALT 35222 uses eqriv 2735 (which requires a biconditional of membership
of
a given setvar variable). Note that abbii 2808, through its closed form
abbi1 2806, is proved from eqrdv 2736, which is the deduction form of
eqriv 2735. In the other direction, velpw 4538 and velsn 4577 are proved from the
definitions of powerclass and singleton using elabg 3607, which is a version
of abbii 2808 suited for membership characterizations.
(Contributed by BJ,
14-Apr-2024.) (Proof modification is discouraged.)
(New usage is discouraged.) |
