| Description: Alternate proof of pw0 4833.
The proofs have a similar structure: pw0 4833
uses the definitions of powerclass and singleton as class abstractions,
whereas bj-pw0ALT 38155 uses characterizations of their elements.
Both proofs
then use transitivity of a congruence relation (equality for pw0 4833
and
biconditional for bj-pw0ALT 38155) to translate the property ss0b 4354
into the
wanted result. To translate a biconditional into a class equality, pw0 4833
uses abbii 2830 (which yields an equality of class
abstractions), while
bj-pw0ALT 38155 uses eqriv 2760 (which requires a biconditional of membership
of
a given setvar variable). Theorem abbii 2830, through its closed form
abbi 2828, is proved from eqrdv 2761, which is the deduction form of eqriv 2760.
In the other direction, velpw 4565 and velsn 4600 are proved from the
definitions of powerclass and singleton using elabg 3633, which is a version
of abbii 2830 suited for membership characterizations.
(Contributed by BJ,
14-Apr-2024.) (Proof modification is discouraged.)
(New usage is discouraged.) |
