Theorem partsuc

Description: Property of the partition. (Contributed by Peter Mazsa, 20-Sep-2024)

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

Proof

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