Theorem neq0r 3429

Description: An inhabited class is nonempty. See n0rf 3427 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 3428	. 2 |- ( E. x x e. A -> A =/= (/) )
2	1	neneqd 2361	1 |- ( E. x x e. A -> -. A = (/) )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1348   E.wex 1485   e. wcel 2141   (/)c0 3414

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-v 2732  df-dif 3123  df-nul 3415

This theorem is referenced by:  exmidsssn  4188  fzn  9998