Theorem nel0 4304

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 4299	. 2 ⊢ (𝐴 = ∅ ↔ ∀𝑥 ¬ 𝑥 ∈ 𝐴)
2		nel0.1	. 2 ⊢ ¬ 𝑥 ∈ 𝐴
3	1, 2	mpgbir 1831	1 ⊢ 𝐴 = ∅

This proof depends on syntax axioms:  ¬ wn 3   = wceq 1569   ∈ wcel 2146  ∅c0 4280

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-9 2156  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2101  df-clab 2742  df-cleq 2755  df-dif 4049  df-nul 4281

This theorem is used by:  uni0  4898  iun0  5022  0iun  5023  0xp  5754  xp0  5755  cnv0  5865  cnv0OLD  5866  dm0  5906  fzouzdisj  13840  bj-ccinftydisj  38274  finxp0  38454  stoweidlem44  47193