| Description: Alternate proof of pw0 4837.
The proofs have a similar structure: pw0 4837
uses the definitions of powerclass and singleton as class abstractions,
whereas bj-pw0ALT 37015 uses characterizations of their elements.
Both proofs
then use transitivity of a congruence relation (equality for pw0 4837
and
biconditional for bj-pw0ALT 37015) to translate the property ss0b 4424
into the
wanted result. To translate a biconditional into a class equality, pw0 4837
uses abbii 2812 (which yields an equality of class
abstractions), while
bj-pw0ALT 37015 uses eqriv 2737 (which requires a biconditional of membership
of
a given setvar variable). Note that abbii 2812, through its closed form
abbi 2810, is proved from eqrdv 2738, which is the deduction form of eqriv 2737.
In the other direction, velpw 4627 and velsn 4664 are proved from the
definitions of powerclass and singleton using elabg 3690, which is a version
of abbii 2812 suited for membership characterizations.
(Contributed by BJ,
14-Apr-2024.) (Proof modification is discouraged.)
(New usage is discouraged.) |
