| Description: Alternate proof of pw0 4772.
The proofs have a similar structure: pw0 4772
uses the definitions of powerclass and singleton as class abstractions,
whereas bj-pw0ALT 37884 uses characterizations of their elements.
Both proofs
then use transitivity of a congruence relation (equality for pw0 4772
and
biconditional for bj-pw0ALT 37884) to translate the property ss0b 4350
into the
wanted result. To translate a biconditional into a class equality, pw0 4772
uses abbii 2827 (which yields an equality of class
abstractions), while
bj-pw0ALT 37884 uses eqriv 2757 (which requires a biconditional of membership
of
a given setvar variable). Note that abbii 2827, through its closed form
abbi 2825, is proved from eqrdv 2758, which is the deduction form of eqriv 2757.
In the other direction, velpw 4561 and velsn 4599 are proved from the
definitions of powerclass and singleton using elabg 3629, which is a version
of abbii 2827 suited for membership characterizations.
(Contributed by BJ,
14-Apr-2024.) (Proof modification is discouraged.)
(New usage is discouraged.) |
