Theorem orddif

Description: Ordinal derived from its successor. (Contributed by NM, 20-May-1998)

	Ref	Expression
Assertion	orddif	$$⊢\mathrm{Ord}A\to A=\mathrm{suc}A\setminus \left\{A\right\}$$

Proof

Step	Hyp	Ref	Expression
1		orddisj	$$⊢\mathrm{Ord}A\to A\cap \left\{A\right\}=\varnothing$$
2		disj3	$$⊢A\cap \left\{A\right\}=\varnothing \leftrightarrow A=A\setminus \left\{A\right\}$$
3		df-suc	$$⊢\mathrm{suc}A=A\cup \left\{A\right\}$$
4	3	difeq1i	$$⊢\mathrm{suc}A\setminus \left\{A\right\}=\left(A\cup \left\{A\right\}\right)\setminus \left\{A\right\}$$
5		difun2	$$⊢\left(A\cup \left\{A\right\}\right)\setminus \left\{A\right\}=A\setminus \left\{A\right\}$$
6	4 5	eqtri	$$⊢\mathrm{suc}A\setminus \left\{A\right\}=A\setminus \left\{A\right\}$$
7	6	eqeq2i	$$⊢A=\mathrm{suc}A\setminus \left\{A\right\}\leftrightarrow A=A\setminus \left\{A\right\}$$
8	2 7	bitr4i	$$⊢A\cap \left\{A\right\}=\varnothing \leftrightarrow A=\mathrm{suc}A\setminus \left\{A\right\}$$
9	1 8	sylib	$$⊢\mathrm{Ord}A\to A=\mathrm{suc}A\setminus \left\{A\right\}$$