Theorem rankeq0

Description: A set is empty iff its rank is empty. (Contributed by NM, 18-Sep-2006) (Revised by Mario Carneiro, 17-Nov-2014)

	Ref	Expression
Hypothesis	rankeq0.1	$$⊢A\in \mathrm{V}$$
Assertion	rankeq0	$$⊢A=\varnothing \leftrightarrow \mathrm{rank}\left(A\right)=\varnothing$$

Proof

Step	Hyp	Ref	Expression
1		rankeq0.1	$$⊢A\in \mathrm{V}$$
2		unir1	$$⊢\bigcup R_1\left[\mathrm{On}\right]=\mathrm{V}$$
3	1 2	eleqtrri	$$⊢A\in \bigcup R_1\left[\mathrm{On}\right]$$
4		rankeq0b	$$⊢A\in \bigcup R_1\left[\mathrm{On}\right]\to \left(A=\varnothing \leftrightarrow \mathrm{rank}\left(A\right)=\varnothing \right)$$
5	3 4	ax-mp	$$⊢A=\varnothing \leftrightarrow \mathrm{rank}\left(A\right)=\varnothing$$