Theorem oe0m0

Description: Ordinal exponentiation with zero base and zero exponent. Proposition 8.31 of TakeutiZaring p. 67. (Contributed by NM, 31-Dec-2004)

	Ref	Expression
Assertion	oe0m0	$$⊢\varnothing {\uparrow }_{𝑜}\varnothing =1_{𝑜}$$

Proof

Step	Hyp	Ref	Expression
1		0elon	$$⊢\varnothing \in \mathrm{On}$$
2		oe0m	$$⊢\varnothing \in \mathrm{On}\to \varnothing {\uparrow }_{𝑜}\varnothing =1_{𝑜}\setminus \varnothing$$
3	1 2	ax-mp	$$⊢\varnothing {\uparrow }_{𝑜}\varnothing =1_{𝑜}\setminus \varnothing$$
4		dif0	$$⊢1_{𝑜}\setminus \varnothing =1_{𝑜}$$
5	3 4	eqtri	$$⊢\varnothing {\uparrow }_{𝑜}\varnothing =1_{𝑜}$$