Theorem oe0m

Description: Value of zero raised to an ordinal. (Contributed by NM, 31-Dec-2004) (Revised by Mario Carneiro, 8-Sep-2013)

	Ref	Expression
Assertion	oe0m	⊢ ( 𝐴 ∈ On → ( ∅ ↑o 𝐴 ) = ( 1o ∖ 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		0elon	⊢ ∅ ∈ On
2		oev	⊢ ( ( ∅ ∈ On ∧ 𝐴 ∈ On ) → ( ∅ ↑o 𝐴 ) = if ( ∅ = ∅ , ( 1o ∖ 𝐴 ) , ( rec ( ( 𝑥 ∈ V ↦ ( 𝑥 ·o ∅ ) ) , 1o ) ‘ 𝐴 ) ) )
3	1 2	mpan	⊢ ( 𝐴 ∈ On → ( ∅ ↑o 𝐴 ) = if ( ∅ = ∅ , ( 1o ∖ 𝐴 ) , ( rec ( ( 𝑥 ∈ V ↦ ( 𝑥 ·o ∅ ) ) , 1o ) ‘ 𝐴 ) ) )
4		eqid	⊢ ∅ = ∅
5	4	iftruei	⊢ if ( ∅ = ∅ , ( 1o ∖ 𝐴 ) , ( rec ( ( 𝑥 ∈ V ↦ ( 𝑥 ·o ∅ ) ) , 1o ) ‘ 𝐴 ) ) = ( 1o ∖ 𝐴 )
6	3 5	eqtrdi	⊢ ( 𝐴 ∈ On → ( ∅ ↑o 𝐴 ) = ( 1o ∖ 𝐴 ) )