Theorem nel0 4199

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 ∈ 𝐴

Assertion

Ref	Expression
nel0	⊢ 𝐴 = ∅

Distinct variable group:   𝑥,𝐴

Proof of Theorem nel0

Step	Hyp	Ref	Expression
1		eq0 4196	. 2 ⊢ (𝐴 = ∅ ↔ ∀𝑥 ¬ 𝑥 ∈ 𝐴)
2		nel0.1	. 2 ⊢ ¬ 𝑥 ∈ 𝐴
3	1, 2	mpgbir 1762	1 ⊢ 𝐴 = ∅

Syntax hints:  ¬ wn 3   = wceq 1507   ∈ wcel 2050  ∅c0 4180

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-11 2093  ax-12 2106  ax-ext 2750

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-dif 3834  df-nul 4181

This theorem is referenced by:  iun0  4852  0iun  4853  0xp  5500  dm0  5638  cnv0  5841  fzouzdisj  12891  bj-ccinftydisj  33964  finxp0  34113  stoweidlem44  41761