Theorem recsexd

Description: A non-zero surreal has a reciprocal. (Contributed by Scott Fenton, 16-Mar-2025)

	Ref	Expression
Hypotheses	recsexd.1	$$⊢\phi \to A\in \mathrm{No}$$
	recsexd.2	$$⊢\phi \to A\ne 0_s$$
Assertion	recsexd	$$⊢\phi \to \exists x\in \mathrm{No}A{\cdot }_{s}x=1_s$$

Proof

Step	Hyp	Ref	Expression
1		recsexd.1	$$⊢\phi \to A\in \mathrm{No}$$
2		recsexd.2	$$⊢\phi \to A\ne 0_s$$
3		recsex	$$⊢\left(A\in \mathrm{No}\wedge A\ne 0_s\right)\to \exists x\in \mathrm{No}A{\cdot }_{s}x=1_s$$
4	1 2 3	syl2anc	$$⊢\phi \to \exists x\in \mathrm{No}A{\cdot }_{s}x=1_s$$