| Description: Alternate proof of pw0 4742.
The proofs have a similar structure: pw0 4742
uses the definitions of powerclass and singleton as class abstractions,
whereas bj-pw0ALT 35149 uses characterizations of their elements.
Both proofs
then use transitivity of a congruence relation (equality for pw0 4742
and
biconditional for bj-pw0ALT 35149) to translate the property ss0b 4328
into the
wanted result. To translate a biconditional into a class equality, pw0 4742
uses abbii 2809 (which yields an equality of class
abstractions), while
bj-pw0ALT 35149 uses eqriv 2735 (which requires a biconditional of membership
of
a given setvar variable). Note that abbii 2809, through its closed form
abbi1 2807, is proved from eqrdv 2736, which is the deduction form of
eqriv 2735. In the other direction, velpw 4535 and velsn 4574 are proved from the
definitions of powerclass and singleton using elabg 3600, which is a version
of abbii 2809 suited for membership characterizations.
(Contributed by BJ,
14-Apr-2024.) (Proof modification is discouraged.)
(New usage is discouraged.) |
