Theorem bj-ab0 32094

Description: The class of sets verifying a falsity is the empty set (closed form of abf 3930). (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		ax-5 1827	. . 3 ⊢ (∀𝑥 ¬ 𝜑 → ∀𝑦∀𝑥 ¬ 𝜑)
2		bj-stdpc4v 31942	. . . . 5 ⊢ (∀𝑥 ¬ 𝜑 → [𝑦 / 𝑥] ¬ 𝜑)
3		sbn 2379	. . . . 5 ⊢ ([𝑦 / 𝑥] ¬ 𝜑 ↔ ¬ [𝑦 / 𝑥]𝜑)
4	2, 3	sylib 207	. . . 4 ⊢ (∀𝑥 ¬ 𝜑 → ¬ [𝑦 / 𝑥]𝜑)
5		df-clab 2597	. . . 4 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ [𝑦 / 𝑥]𝜑)
6	4, 5	sylnibr 318	. . 3 ⊢ (∀𝑥 ¬ 𝜑 → ¬ 𝑦 ∈ {𝑥 ∣ 𝜑})
7	1, 6	alrimih 1741	. 2 ⊢ (∀𝑥 ¬ 𝜑 → ∀𝑦 ¬ 𝑦 ∈ {𝑥 ∣ 𝜑})
8		eq0 3888	. 2 ⊢ ({𝑥 ∣ 𝜑} = ∅ ↔ ∀𝑦 ¬ 𝑦 ∈ {𝑥 ∣ 𝜑})
9	7, 8	sylibr 223	1 ⊢ (∀𝑥 ¬ 𝜑 → {𝑥 ∣ 𝜑} = ∅)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1473   = wceq 1475  [wsb 1867   ∈ wcel 1977  {cab 2596  ∅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-abf  32095  bj-csbprc  32096