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 → suc 𝐴 = suc 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		id	⊢ ( 𝐴 = 𝐵 → 𝐴 = 𝐵 )
2		sneq	⊢ ( 𝐴 = 𝐵 → { 𝐴 } = { 𝐵 } )
3	1 2	uneq12d	⊢ ( 𝐴 = 𝐵 → ( 𝐴 ∪ { 𝐴 } ) = ( 𝐵 ∪ { 𝐵 } ) )
4		df-suc	⊢ suc 𝐴 = ( 𝐴 ∪ { 𝐴 } )
5		df-suc	⊢ suc 𝐵 = ( 𝐵 ∪ { 𝐵 } )
6	3 4 5	3eqtr4g	⊢ ( 𝐴 = 𝐵 → suc 𝐴 = suc 𝐵 )