Theorem sucel

Description: Membership of a successor in another class. (Contributed by NM, 29-Jun-2004)

	Ref	Expression
Assertion	sucel	$$⊢\mathrm{suc}A\in B\leftrightarrow \exists x\in B\forall y\left(y\in x\leftrightarrow \left(y\in A\vee y=A\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		risset	$$⊢\mathrm{suc}A\in B\leftrightarrow \exists x\in Bx=\mathrm{suc}A$$
2		dfcleq	$$⊢x=\mathrm{suc}A\leftrightarrow \forall y\left(y\in x\leftrightarrow y\in \mathrm{suc}A\right)$$
3		vex	$$⊢y\in \mathrm{V}$$
4	3	elsuc	$$⊢y\in \mathrm{suc}A\leftrightarrow \left(y\in A\vee y=A\right)$$
5	4	bibi2i	$$⊢\left(y\in x\leftrightarrow y\in \mathrm{suc}A\right)\leftrightarrow \left(y\in x\leftrightarrow \left(y\in A\vee y=A\right)\right)$$
6	5	albii	$$⊢\forall y\left(y\in x\leftrightarrow y\in \mathrm{suc}A\right)\leftrightarrow \forall y\left(y\in x\leftrightarrow \left(y\in A\vee y=A\right)\right)$$
7	2 6	bitri	$$⊢x=\mathrm{suc}A\leftrightarrow \forall y\left(y\in x\leftrightarrow \left(y\in A\vee y=A\right)\right)$$
8	7	rexbii	$$⊢\exists x\in Bx=\mathrm{suc}A\leftrightarrow \exists x\in B\forall y\left(y\in x\leftrightarrow \left(y\in A\vee y=A\right)\right)$$
9	1 8	bitri	$$⊢\mathrm{suc}A\in B\leftrightarrow \exists x\in B\forall y\left(y\in x\leftrightarrow \left(y\in A\vee y=A\right)\right)$$