Theorem spr0el

Description: The empty set is not an unordered pair over any set V . (Contributed by AV, 21-Nov-2021)

	Ref	Expression
Assertion	spr0el	$$⊢\varnothing \notin \mathrm{Pairs}\left(V\right)$$

Proof

Step	Hyp	Ref	Expression
1		spr0nelg	$$⊢\varnothing \notin \left\{p|\exists a\exists bp=\left\{a,b\right\}\right\}$$
2		sprssspr	$$⊢\mathrm{Pairs}\left(V\right)\subseteq \left\{p|\exists a\exists bp=\left\{a,b\right\}\right\}$$
3	2	sseli	$$⊢\varnothing \in \mathrm{Pairs}\left(V\right)\to \varnothing \in \left\{p|\exists a\exists bp=\left\{a,b\right\}\right\}$$
4	3	con3i	$$⊢\neg \varnothing \in \left\{p|\exists a\exists bp=\left\{a,b\right\}\right\}\to \neg \varnothing \in \mathrm{Pairs}\left(V\right)$$
5		df-nel	$$⊢\varnothing \notin \left\{p|\exists a\exists bp=\left\{a,b\right\}\right\}\leftrightarrow \neg \varnothing \in \left\{p|\exists a\exists bp=\left\{a,b\right\}\right\}$$
6		df-nel	$$⊢\varnothing \notin \mathrm{Pairs}\left(V\right)\leftrightarrow \neg \varnothing \in \mathrm{Pairs}\left(V\right)$$
7	4 5 6	3imtr4i	$$⊢\varnothing \notin \left\{p|\exists a\exists bp=\left\{a,b\right\}\right\}\to \varnothing \notin \mathrm{Pairs}\left(V\right)$$
8	1 7	ax-mp	$$⊢\varnothing \notin \mathrm{Pairs}\left(V\right)$$