Theorem nel0 3472

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 3469	. 2 |- ( A = (/) <-> A. x -. x e. A )
2		nel0.1	. 2 |- -. x e. A
3	1, 2	mpgbir 1467	1 |- A = (/)

Syntax hints:   -. wn 3   = wceq 1364   e. wcel 2167   (/)c0 3450

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-dif 3159  df-nul 3451

This theorem is referenced by: (None)