| Description: Alternate proof of pw0 4761.
The proofs have a similar structure: pw0 4761
uses the definitions of powerclass and singleton as class abstractions,
whereas bj-pw0ALT 37093 uses characterizations of their elements.
Both proofs
then use transitivity of a congruence relation (equality for pw0 4761
and
biconditional for bj-pw0ALT 37093) to translate the property ss0b 4348
into the
wanted result. To translate a biconditional into a class equality, pw0 4761
uses abbii 2798 (which yields an equality of class
abstractions), while
bj-pw0ALT 37093 uses eqriv 2728 (which requires a biconditional of membership
of
a given setvar variable). Note that abbii 2798, through its closed form
abbi 2796, is proved from eqrdv 2729, which is the deduction form of eqriv 2728.
In the other direction, velpw 4552 and velsn 4589 are proved from the
definitions of powerclass and singleton using elabg 3627, which is a version
of abbii 2798 suited for membership characterizations.
(Contributed by BJ,
14-Apr-2024.) (Proof modification is discouraged.)
(New usage is discouraged.) |
