Theorem bj-nel0 32128

Description: From the general negation of membership in 𝐴, infer that 𝐴 is the empty set. [Could shorten 0xp 5122?] (Contributed by BJ, 6-Oct-2018.)

Hypothesis

Ref	Expression
bj-nel0.1	⊢ ¬ 𝑥 ∈ 𝐴

Assertion

Ref	Expression
bj-nel0	⊢ 𝐴 = ∅

Distinct variable group:   𝑥,𝐴

Proof of Theorem bj-nel0

Step	Hyp	Ref	Expression
1		eq0 3888	. 2 ⊢ (𝐴 = ∅ ↔ ∀𝑥 ¬ 𝑥 ∈ 𝐴)
2		bj-nel0.1	. 2 ⊢ ¬ 𝑥 ∈ 𝐴
3	1, 2	mpgbir 1717	1 ⊢ 𝐴 = ∅

Syntax hints:  ¬ wn 3   = wceq 1475   ∈ wcel 1977  ∅c0 3874

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-dif 3543  df-nul 3875

This theorem is referenced by:  bj-ccinftydisj  32277