Theorem nel0 4295

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 4291	. 2 ⊢ (𝐴 = ∅ ↔ ∀𝑥 ¬ 𝑥 ∈ 𝐴)
2		nel0.1	. 2 ⊢ ¬ 𝑥 ∈ 𝐴
3	1, 2	mpgbir 1801	1 ⊢ 𝐴 = ∅

Syntax hints:  ¬ wn 3   = wceq 1542   ∈ wcel 2114  ∅c0 4274

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-dif 3893  df-nul 4275

This theorem is referenced by:  uni0  4879  iun0  5005  0iun  5006  0xp  5724  xp0  5725  dm0  5870  cnv0  6098  cnv0OLD  6099  fzouzdisj  13644  bj-ccinftydisj  37546  finxp0  37724  stoweidlem44  46493