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Metamath Proof Explorer

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 = 1 𝑜 ∖ A

Proof

Step Hyp Ref Expression
1 0elon ⊢ ∅ ∈ On
2 oev ⊢ ∅ ∈ On ∧ A ∈ On → ∅ ↑ 𝑜 A = if ∅ = ∅ 1 𝑜 ∖ A rec ⁡ x ∈ V ⟼ x ⋅ 𝑜 ∅ 1 𝑜 ⁡ A
3 1 2 mpan ⊢ A ∈ On → ∅ ↑ 𝑜 A = if ∅ = ∅ 1 𝑜 ∖ A rec ⁡ x ∈ V ⟼ x ⋅ 𝑜 ∅ 1 𝑜 ⁡ A
4 eqid ⊢ ∅ = ∅
5 4 iftruei ⊢ if ∅ = ∅ 1 𝑜 ∖ A rec ⁡ x ∈ V ⟼ x ⋅ 𝑜 ∅ 1 𝑜 ⁡ A = 1 𝑜 ∖ A
6 3 5 eqtrdi ⊢ A ∈ On → ∅ ↑ 𝑜 A = 1 𝑜 ∖ A