| Description: Alternate proof of pw0 4776.
The proofs have a similar structure: pw0 4776
uses the definitions of powerclass and singleton as class abstractions,
whereas bj-pw0ALT 37037 uses characterizations of their elements.
Both proofs
then use transitivity of a congruence relation (equality for pw0 4776
and
biconditional for bj-pw0ALT 37037) to translate the property ss0b 4364
into the
wanted result. To translate a biconditional into a class equality, pw0 4776
uses abbii 2796 (which yields an equality of class
abstractions), while
bj-pw0ALT 37037 uses eqriv 2726 (which requires a biconditional of membership
of
a given setvar variable). Note that abbii 2796, through its closed form
abbi 2794, is proved from eqrdv 2727, which is the deduction form of eqriv 2726.
In the other direction, velpw 4568 and velsn 4605 are proved from the
definitions of powerclass and singleton using elabg 3643, which is a version
of abbii 2796 suited for membership characterizations.
(Contributed by BJ,
14-Apr-2024.) (Proof modification is discouraged.)
(New usage is discouraged.) |
