Theorem hash1elsn

Description: A set of size 1 with a known element is the singleton of that element. (Contributed by Rohan Ridenour, 3-Aug-2023)

	Ref	Expression
Hypotheses	hash1elsn.1	$$⊢\phi \to \left|A\right|=1$$
	hash1elsn.2	$$⊢\phi \to B\in A$$
	hash1elsn.3	$$⊢\phi \to A\in V$$
Assertion	hash1elsn	$$⊢\phi \to A=\left\{B\right\}$$

Proof

Step	Hyp	Ref	Expression
1		hash1elsn.1	$$⊢\phi \to \left|A\right|=1$$
2		hash1elsn.2	$$⊢\phi \to B\in A$$
3		hash1elsn.3	$$⊢\phi \to A\in V$$
4		hashen1	$$⊢A\in V\to \left(\left|A\right|=1\leftrightarrow A\approx 1_{𝑜}\right)$$
5	3 4	syl	$$⊢\phi \to \left(\left|A\right|=1\leftrightarrow A\approx 1_{𝑜}\right)$$
6	1 5	mpbid	$$⊢\phi \to A\approx 1_{𝑜}$$
7		en1	$$⊢A\approx 1_{𝑜}\leftrightarrow \exists xA=\left\{x\right\}$$
8	6 7	sylib	$$⊢\phi \to \exists xA=\left\{x\right\}$$
9		simpr	$$⊢\left(\phi \wedge A=\left\{x\right\}\right)\to A=\left\{x\right\}$$
10	2	adantr	$$⊢\left(\phi \wedge A=\left\{x\right\}\right)\to B\in A$$
11	10 9	eleqtrd	$$⊢\left(\phi \wedge A=\left\{x\right\}\right)\to B\in \left\{x\right\}$$
12		elsni	$$⊢B\in \left\{x\right\}\to B=x$$
13	11 12	syl	$$⊢\left(\phi \wedge A=\left\{x\right\}\right)\to B=x$$
14	13	sneqd	$$⊢\left(\phi \wedge A=\left\{x\right\}\right)\to \left\{B\right\}=\left\{x\right\}$$
15	9 14	eqtr4d	$$⊢\left(\phi \wedge A=\left\{x\right\}\right)\to A=\left\{B\right\}$$
16	8 15	exlimddv	$$⊢\phi \to A=\left\{B\right\}$$