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Metamath Proof Explorer

Theorem nnasuc

Description: Addition with successor. Theorem 4I(A2) of Enderton p. 79. (Contributed by NM, 20-Sep-1995) (Revised by Mario Carneiro, 14-Nov-2014)

Ref Expression
Assertion nnasuc ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) → ( 𝐴 +o suc 𝐵 ) = suc ( 𝐴 +o 𝐵 ) )

Proof

Step Hyp Ref Expression
1 nnon ⊢ ( 𝐴 ∈ ω → 𝐴 ∈ On )
2 onasuc ⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ ω ) → ( 𝐴 +o suc 𝐵 ) = suc ( 𝐴 +o 𝐵 ) )
3 1 2 sylan ⊢ ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) → ( 𝐴 +o suc 𝐵 ) = suc ( 𝐴 +o 𝐵 ) )