Theorem nel0 4309

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 4304	. 2 ⊢ (𝐴 = ∅ ↔ ∀𝑥 ¬ 𝑥 ∈ 𝐴)
2		nel0.1	. 2 ⊢ ¬ 𝑥 ∈ 𝐴
3	1, 2	mpgbir 1832	1 ⊢ 𝐴 = ∅

This proof depends on syntax axioms:  ¬ wn 3   = wceq 1570   ∈ wcel 2146  ∅c0 4286

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-9 2156  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-dif 3909  df-nul 4287

This theorem is used by:  uni0  4903  iun0  5028  0iun  5029  0xp  5762  xp0  5763  cnv0  5871  cnv0OLD  5872  dm0  5912  fzouzdisj  13743  bj-ccinftydisj  37916  finxp0  38096  stoweidlem44  46818