| Description: Alternate proof of pw0 4750.
The proofs have a similar structure: pw0 4750
uses the definitions of powerclass and singleton as class abstractions,
whereas bj-pw0ALT 37403 uses characterizations of their elements.
Both proofs
then use transitivity of a congruence relation (equality for pw0 4750
and
biconditional for bj-pw0ALT 37403) to translate the property ss0b 4336
into the
wanted result. To translate a biconditional into a class equality, pw0 4750
uses abbii 2807 (which yields an equality of class
abstractions), while
bj-pw0ALT 37403 uses eqriv 2737 (which requires a biconditional of membership
of
a given setvar variable). Note that abbii 2807, through its closed form
abbi 2805, is proved from eqrdv 2738, which is the deduction form of eqriv 2737.
In the other direction, velpw 4541 and velsn 4578 are proved from the
definitions of powerclass and singleton using elabg 3621, which is a version
of abbii 2807 suited for membership characterizations.
(Contributed by BJ,
14-Apr-2024.) (Proof modification is discouraged.)
(New usage is discouraged.) |
