Theorem neq0r 3506

Description: An inhabited class is nonempty. See n0rf 3504 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 3505	. 2 |- ( E. x x e. A -> A =/= (/) )
2	1	neneqd 2421	1 |- ( E. x x e. A -> -. A = (/) )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1395   E.wex 1538   e. wcel 2200   (/)c0 3491

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-v 2801  df-dif 3199  df-nul 3492

This theorem is referenced by:  exmidsssn  4285  fzn  10234