Theorem bj-ab0 36850

Description: The class of sets verifying a falsity is the empty set (closed form of abf 4388). (Contributed by BJ, 24-Jul-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-ab0	⊢ (∀𝑥 ¬ 𝜑 → {𝑥 ∣ 𝜑} = ∅)

Proof of Theorem bj-ab0

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		stdpc4 2067	. . . 4 ⊢ (∀𝑥 ¬ 𝜑 → [𝑦 / 𝑥] ¬ 𝜑)
2		sbn1 2106	. . . 4 ⊢ ([𝑦 / 𝑥] ¬ 𝜑 → ¬ [𝑦 / 𝑥]𝜑)
3	1, 2	syl 17	. . 3 ⊢ (∀𝑥 ¬ 𝜑 → ¬ [𝑦 / 𝑥]𝜑)
4		df-clab 2713	. . 3 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ [𝑦 / 𝑥]𝜑)
5	3, 4	sylnibr 329	. 2 ⊢ (∀𝑥 ¬ 𝜑 → ¬ 𝑦 ∈ {𝑥 ∣ 𝜑})
6	5	eq0rdv 4389	1 ⊢ (∀𝑥 ¬ 𝜑 → {𝑥 ∣ 𝜑} = ∅)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1537   = wceq 1539  [wsb 2063   ∈ wcel 2107  {cab 2712  ∅c0 4315

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-9 2117  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2713  df-cleq 2726  df-dif 3936  df-nul 4316

This theorem is referenced by:  bj-abf  36851  bj-csbprc  36852