Theorem 0g0

Description: The identity element function evaluates to the empty set on an empty structure. (Contributed by Stefan O'Rear, 2-Oct-2015)

	Ref	Expression
Assertion	0g0	$$⊢\varnothing =0_{\varnothing }$$

Proof

Step	Hyp	Ref	Expression
1		base0	$$⊢\varnothing ={\mathrm{Base}}_{\varnothing }$$
2		eqid	$$⊢{+}_{\varnothing }={+}_{\varnothing }$$
3		eqid	$$⊢0_{\varnothing }=0_{\varnothing }$$
4	1 2 3	grpidval	$$⊢0_{\varnothing }=\left(\iota e|\left(e\in \varnothing \wedge \forall x\in \varnothing \left(e{+}_{\varnothing }x=x\wedge x{+}_{\varnothing }e=x\right)\right)\right)$$
5		noel	$$⊢\neg e\in \varnothing$$
6	5	intnanr	$$⊢\neg \left(e\in \varnothing \wedge \forall x\in \varnothing \left(e{+}_{\varnothing }x=x\wedge x{+}_{\varnothing }e=x\right)\right)$$
7	6	nex	$$⊢\neg \exists e\left(e\in \varnothing \wedge \forall x\in \varnothing \left(e{+}_{\varnothing }x=x\wedge x{+}_{\varnothing }e=x\right)\right)$$
8		euex	$$⊢\exists !e\left(e\in \varnothing \wedge \forall x\in \varnothing \left(e{+}_{\varnothing }x=x\wedge x{+}_{\varnothing }e=x\right)\right)\to \exists e\left(e\in \varnothing \wedge \forall x\in \varnothing \left(e{+}_{\varnothing }x=x\wedge x{+}_{\varnothing }e=x\right)\right)$$
9	7 8	mto	$$⊢\neg \exists !e\left(e\in \varnothing \wedge \forall x\in \varnothing \left(e{+}_{\varnothing }x=x\wedge x{+}_{\varnothing }e=x\right)\right)$$
10		iotanul	$$⊢\neg \exists !e\left(e\in \varnothing \wedge \forall x\in \varnothing \left(e{+}_{\varnothing }x=x\wedge x{+}_{\varnothing }e=x\right)\right)\to \left(\iota e|\left(e\in \varnothing \wedge \forall x\in \varnothing \left(e{+}_{\varnothing }x=x\wedge x{+}_{\varnothing }e=x\right)\right)\right)=\varnothing$$
11	9 10	ax-mp	$$⊢\left(\iota e|\left(e\in \varnothing \wedge \forall x\in \varnothing \left(e{+}_{\varnothing }x=x\wedge x{+}_{\varnothing }e=x\right)\right)\right)=\varnothing$$
12	4 11	eqtr2i	$$⊢\varnothing =0_{\varnothing }$$