Theorem suceq

Description: Equality of successors. (Contributed by NM, 30-Aug-1993) (Proof shortened by Andrew Salmon, 25-Jul-2011)

	Ref	Expression
Assertion	suceq	$$⊢A=B\to \mathrm{suc}A=\mathrm{suc}B$$

Proof

Step	Hyp	Ref	Expression
1		id	$$⊢A=B\to A=B$$
2		sneq	$$⊢A=B\to \left\{A\right\}=\left\{B\right\}$$
3	1 2	uneq12d	$$⊢A=B\to A\cup \left\{A\right\}=B\cup \left\{B\right\}$$
4		df-suc	$$⊢\mathrm{suc}A=A\cup \left\{A\right\}$$
5		df-suc	$$⊢\mathrm{suc}B=B\cup \left\{B\right\}$$
6	3 4 5	3eqtr4g	$$⊢A=B\to \mathrm{suc}A=\mathrm{suc}B$$