Theorem nel0 3331

Description: From the general negation of membership in A, infer that A is the empty set. (Contributed by BJ, 6-Oct-2018.)

Hypothesis

Ref	Expression
nel0.1	|- -. x e. A

Assertion

Ref	Expression
nel0	|- A = (/)

Distinct variable group:   x, A

Proof of Theorem nel0

Step	Hyp	Ref	Expression
1		eq0 3328	. 2 |- ( A = (/) <-> A. x -. x e. A )
2		nel0.1	. 2 |- -. x e. A
3	1, 2	mpgbir 1397	1 |- A = (/)

Syntax hints:   -. wn 3   = wceq 1299   e. wcel 1448   (/)c0 3310

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082

This theorem depends on definitions:  df-bi 116  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-v 2643  df-dif 3023  df-nul 3311

This theorem is referenced by: (None)