Theorem naddov

Description: The value of natural addition. (Contributed by Scott Fenton, 26-Aug-2024)

	Ref	Expression
Assertion	naddov	⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( 𝐴 +no 𝐵 ) = ∩ { 𝑥 ∈ On ∣ ( ( +no “ ( { 𝐴 } × 𝐵 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝐴 × { 𝐵 } ) ) ⊆ 𝑥 ) } )

Proof

Step	Hyp	Ref	Expression
1		naddcllem	⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( ( 𝐴 +no 𝐵 ) ∈ On ∧ ( 𝐴 +no 𝐵 ) = ∩ { 𝑥 ∈ On ∣ ( ( +no “ ( { 𝐴 } × 𝐵 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝐴 × { 𝐵 } ) ) ⊆ 𝑥 ) } ) )
2	1	simprd	⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( 𝐴 +no 𝐵 ) = ∩ { 𝑥 ∈ On ∣ ( ( +no “ ( { 𝐴 } × 𝐵 ) ) ⊆ 𝑥 ∧ ( +no “ ( 𝐴 × { 𝐵 } ) ) ⊆ 𝑥 ) } )