Theorem partsuc2

Description: Property of the partition. (Contributed by Peter Mazsa, 24-Jul-2024)

	Ref	Expression
Assertion	partsuc2	Could not format assertion : No typesetting found for |- ( ( ( R |` ( A u. { A } ) ) \ ( R |` { A } ) ) Part ( ( A u. { A } ) \ { A } ) <-> ( R |` A ) Part A ) with typecode |-

Proof

Step	Hyp	Ref	Expression
1		ressucdifsn2	$$⊢\left({R\upharpoonright }_{\left(A\cup \left\{A\right\}\right)}\right)\setminus \left({R\upharpoonright }_{\left\{A\right\}}\right)={R\upharpoonright }_A$$
2		sucdifsn2	$$⊢\left(A\cup \left\{A\right\}\right)\setminus \left\{A\right\}=A$$
3		parteq12	Could not format ( ( ( ( R |` ( A u. { A } ) ) \ ( R |` { A } ) ) = ( R |` A ) /\ ( ( A u. { A } ) \ { A } ) = A ) -> ( ( ( R |` ( A u. { A } ) ) \ ( R |` { A } ) ) Part ( ( A u. { A } ) \ { A } ) <-> ( R |` A ) Part A ) ) : No typesetting found for |- ( ( ( ( R |` ( A u. { A } ) ) \ ( R |` { A } ) ) = ( R |` A ) /\ ( ( A u. { A } ) \ { A } ) = A ) -> ( ( ( R |` ( A u. { A } ) ) \ ( R |` { A } ) ) Part ( ( A u. { A } ) \ { A } ) <-> ( R |` A ) Part A ) ) with typecode |-
4	1 2 3	mp2an	Could not format ( ( ( R |` ( A u. { A } ) ) \ ( R |` { A } ) ) Part ( ( A u. { A } ) \ { A } ) <-> ( R |` A ) Part A ) : No typesetting found for |- ( ( ( R |` ( A u. { A } ) ) \ ( R |` { A } ) ) Part ( ( A u. { A } ) \ { A } ) <-> ( R |` A ) Part A ) with typecode |-