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Theorem bj-ab0 32886
Description: The class of sets verifying a falsity is the empty set (closed form of abf 3976). (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 1838 . . 3 (∀𝑥 ¬ 𝜑 → ∀𝑦𝑥 ¬ 𝜑)
2 bj-stdpc4v 32738 . . . . 5 (∀𝑥 ¬ 𝜑 → [𝑦 / 𝑥] ¬ 𝜑)
3 sbn 2390 . . . . 5 ([𝑦 / 𝑥] ¬ 𝜑 ↔ ¬ [𝑦 / 𝑥]𝜑)
42, 3sylib 208 . . . 4 (∀𝑥 ¬ 𝜑 → ¬ [𝑦 / 𝑥]𝜑)
5 df-clab 2608 . . . 4 (𝑦 ∈ {𝑥𝜑} ↔ [𝑦 / 𝑥]𝜑)
64, 5sylnibr 319 . . 3 (∀𝑥 ¬ 𝜑 → ¬ 𝑦 ∈ {𝑥𝜑})
71, 6alrimih 1750 . 2 (∀𝑥 ¬ 𝜑 → ∀𝑦 ¬ 𝑦 ∈ {𝑥𝜑})
8 eq0 3927 . 2 ({𝑥𝜑} = ∅ ↔ ∀𝑦 ¬ 𝑦 ∈ {𝑥𝜑})
97, 8sylibr 224 1 (∀𝑥 ¬ 𝜑 → {𝑥𝜑} = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1480   = wceq 1482  [wsb 1879  wcel 1989  {cab 2607  c0 3913
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-v 3200  df-dif 3575  df-nul 3914
This theorem is referenced by:  bj-abf  32887  bj-csbprc  32888
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