Theorem nel0 3543

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 3540	. 2 |- ( A = (/) <-> A. x -. x e. A )
2		nel0.1	. 2 |- -. x e. A
3	1, 2	mpgbir 1506	1 |- A = (/)

Syntax hints:   -. wn 3   = wceq 1402   e. wcel 2209   (/)c0 3520

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-dif 3222  df-nul 3521

This theorem is referenced by:  fczsupp0  6489