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Theorem bj-ab0 36625
Description: The class of sets verifying a falsity is the empty set (closed form of abf 4401). (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 stdpc4 2064 . . . 4 (∀𝑥 ¬ 𝜑 → [𝑦 / 𝑥] ¬ 𝜑)
2 sbn1 2098 . . . 4 ([𝑦 / 𝑥] ¬ 𝜑 → ¬ [𝑦 / 𝑥]𝜑)
31, 2syl 17 . . 3 (∀𝑥 ¬ 𝜑 → ¬ [𝑦 / 𝑥]𝜑)
4 df-clab 2704 . . 3 (𝑦 ∈ {𝑥𝜑} ↔ [𝑦 / 𝑥]𝜑)
53, 4sylnibr 328 . 2 (∀𝑥 ¬ 𝜑 → ¬ 𝑦 ∈ {𝑥𝜑})
65eq0rdv 4402 1 (∀𝑥 ¬ 𝜑 → {𝑥𝜑} = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1532   = wceq 1534  [wsb 2060  wcel 2099  {cab 2703  c0 4323
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-9 2109  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-dif 3950  df-nul 4324
This theorem is referenced by:  bj-abf  36626  bj-csbprc  36627
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