Theorem bj-ab0 38035

Description: The class of sets verifying a falsity is the empty set (closed form of abf 4367). (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 2107	. . . 4 ⊢ (∀𝑥 ¬ 𝜑 → [𝑦 / 𝑥] ¬ 𝜑)
2		sbn1 2146	. . . 4 ⊢ ([𝑦 / 𝑥] ¬ 𝜑 → ¬ [𝑦 / 𝑥]𝜑)
3	1, 2	syl 18	. . 3 ⊢ (∀𝑥 ¬ 𝜑 → ¬ [𝑦 / 𝑥]𝜑)
4		df-clab 2742	. . 3 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ [𝑦 / 𝑥]𝜑)
5	3, 4	sylnibr 332	. 2 ⊢ (∀𝑥 ¬ 𝜑 → ¬ 𝑦 ∈ {𝑥 ∣ 𝜑})
6	5	eq0rdv 4368	1 ⊢ (∀𝑥 ¬ 𝜑 → {𝑥 ∣ 𝜑} = ∅)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4  ∀wal 1568   = wceq 1570  [wsb 2101   ∈ wcel 2147  {cab 2741  ∅c0 4281

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 1945  ax-6 2002  ax-7 2043  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-dif 4050  df-nul 4282

This theorem is used by:  bj-abf  38036  bj-csbprc  38037