Theorem neq0r 3509

Description: An inhabited class is nonempty. See n0rf 3507 for more discussion. (Contributed by Jim Kingdon, 31-Jul-2018.)

Assertion

Ref	Expression
neq0r	|- ( E. x x e. A -> -. A = (/) )

Distinct variable group:   x, A

Proof of Theorem neq0r

Step	Hyp	Ref	Expression
1		n0r 3508	. 2 |- ( E. x x e. A -> A =/= (/) )
2	1	neneqd 2423	1 |- ( E. x x e. A -> -. A = (/) )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1397   E.wex 1540   e. wcel 2202   (/)c0 3494

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-v 2804  df-dif 3202  df-nul 3495

This theorem is referenced by:  exmidsssn  4292  fzn  10276