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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 → {𝑥𝜑} = ∅)

Proof of Theorem bj-ab0
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 ax-5 1827 . . 3 (∀𝑥 ¬ 𝜑 → ∀𝑦𝑥 ¬ 𝜑)
2 bj-stdpc4v 31942 . . . . 5 (∀𝑥 ¬ 𝜑 → [𝑦 / 𝑥] ¬ 𝜑)
3 sbn 2379 . . . . 5 ([𝑦 / 𝑥] ¬ 𝜑 ↔ ¬ [𝑦 / 𝑥]𝜑)
42, 3sylib 207 . . . 4 (∀𝑥 ¬ 𝜑 → ¬ [𝑦 / 𝑥]𝜑)
5 df-clab 2597 . . . 4 (𝑦 ∈ {𝑥𝜑} ↔ [𝑦 / 𝑥]𝜑)
64, 5sylnibr 318 . . 3 (∀𝑥 ¬ 𝜑 → ¬ 𝑦 ∈ {𝑥𝜑})
71, 6alrimih 1741 . 2 (∀𝑥 ¬ 𝜑 → ∀𝑦 ¬ 𝑦 ∈ {𝑥𝜑})
8 eq0 3888 . 2 ({𝑥𝜑} = ∅ ↔ ∀𝑦 ¬ 𝑦 ∈ {𝑥𝜑})
97, 8sylibr 223 1 (∀𝑥 ¬ 𝜑 → {𝑥𝜑} = ∅)
 Colors of variables: wff setvar class 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
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