Theorem ruALT

Description: Alternate proof of ru , simplified using (indirectly) the Axiom of Regularity ax-reg . (Contributed by Alan Sare, 4-Oct-2008) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Assertion	ruALT	$$⊢\left\{x|x\notin x\right\}\notin \mathrm{V}$$

Proof

Step	Hyp	Ref	Expression
1		vprc	$$⊢\neg \mathrm{V}\in \mathrm{V}$$
2	1	nelir	$$⊢\mathrm{V}\notin \mathrm{V}$$
3		ruv	$$⊢\left\{x|x\notin x\right\}=\mathrm{V}$$
4		neleq1	$$⊢\left\{x|x\notin x\right\}=\mathrm{V}\to \left(\left\{x|x\notin x\right\}\notin \mathrm{V}\leftrightarrow \mathrm{V}\notin \mathrm{V}\right)$$
5	3 4	ax-mp	$$⊢\left\{x|x\notin x\right\}\notin \mathrm{V}\leftrightarrow \mathrm{V}\notin \mathrm{V}$$
6	2 5	mpbir	$$⊢\left\{x|x\notin x\right\}\notin \mathrm{V}$$