Metamath Proof Explorer


Theorem 0nsr

Description: The empty set is not a signed real. (Contributed by NM, 25-Aug-1995) (Revised by Mario Carneiro, 10-Jul-2014) (New usage is discouraged.)

Ref Expression
Assertion 0nsr ¬ 𝑹

Proof

Step Hyp Ref Expression
1 eqid =
2 enrer ~ 𝑹 Er 𝑷 × 𝑷
3 erdm ~ 𝑹 Er 𝑷 × 𝑷 dom ~ 𝑹 = 𝑷 × 𝑷
4 2 3 ax-mp dom ~ 𝑹 = 𝑷 × 𝑷
5 elqsn0 dom ~ 𝑹 = 𝑷 × 𝑷 𝑷 × 𝑷 / ~ 𝑹
6 4 5 mpan 𝑷 × 𝑷 / ~ 𝑹
7 df-nr 𝑹 = 𝑷 × 𝑷 / ~ 𝑹
8 6 7 eleq2s 𝑹
9 8 necon2bi = ¬ 𝑹
10 1 9 ax-mp ¬ 𝑹