Theorem bj-ab0 36942

Description: The class of sets verifying a falsity is the empty set (closed form of abf 4351). (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 2071	. . . 4 ⊢ (∀𝑥 ¬ 𝜑 → [𝑦 / 𝑥] ¬ 𝜑)
2		sbn1 2110	. . . 4 ⊢ ([𝑦 / 𝑥] ¬ 𝜑 → ¬ [𝑦 / 𝑥]𝜑)
3	1, 2	syl 17	. . 3 ⊢ (∀𝑥 ¬ 𝜑 → ¬ [𝑦 / 𝑥]𝜑)
4		df-clab 2710	. . 3 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ [𝑦 / 𝑥]𝜑)
5	3, 4	sylnibr 329	. 2 ⊢ (∀𝑥 ¬ 𝜑 → ¬ 𝑦 ∈ {𝑥 ∣ 𝜑})
6	5	eq0rdv 4352	1 ⊢ (∀𝑥 ¬ 𝜑 → {𝑥 ∣ 𝜑} = ∅)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1539   = wceq 1541  [wsb 2067   ∈ wcel 2111  {cab 2709  ∅c0 4278

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-dif 3900  df-nul 4279

This theorem is referenced by:  bj-abf  36943  bj-csbprc  36944