Theorem 2ndnpr

Description: Value of the second-member function at non-pairs. (Contributed by Thierry Arnoux, 22-Sep-2017)

	Ref	Expression
Assertion	2ndnpr	$$⊢\neg A\in \left(\mathrm{V}\times \mathrm{V}\right)\to 2^{nd}\left(A\right)=\varnothing$$

Proof

Step	Hyp	Ref	Expression
1		2ndval	$$⊢2^{nd}\left(A\right)=\bigcup \mathrm{ran}\left\{A\right\}$$
2		rnsnn0	$$⊢A\in \left(\mathrm{V}\times \mathrm{V}\right)\leftrightarrow \mathrm{ran}\left\{A\right\}\ne \varnothing$$
3	2	biimpri	$$⊢\mathrm{ran}\left\{A\right\}\ne \varnothing \to A\in \left(\mathrm{V}\times \mathrm{V}\right)$$
4	3	necon1bi	$$⊢\neg A\in \left(\mathrm{V}\times \mathrm{V}\right)\to \mathrm{ran}\left\{A\right\}=\varnothing$$
5	4	unieqd	$$⊢\neg A\in \left(\mathrm{V}\times \mathrm{V}\right)\to \bigcup \mathrm{ran}\left\{A\right\}=\bigcup \varnothing$$
6		uni0	$$⊢\bigcup \varnothing =\varnothing$$
7	5 6	eqtrdi	$$⊢\neg A\in \left(\mathrm{V}\times \mathrm{V}\right)\to \bigcup \mathrm{ran}\left\{A\right\}=\varnothing$$
8	1 7	syl5eq	$$⊢\neg A\in \left(\mathrm{V}\times \mathrm{V}\right)\to 2^{nd}\left(A\right)=\varnothing$$