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Theorem bj-ab0 33716
Description: The class of sets verifying a falsity is the empty set (closed form of abf 4236). (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 1869 . . 3 (∀𝑥 ¬ 𝜑 → ∀𝑦𝑥 ¬ 𝜑)
2 stdpc4 2019 . . . . 5 (∀𝑥 ¬ 𝜑 → [𝑦 / 𝑥] ¬ 𝜑)
3 sbn 2214 . . . . 5 ([𝑦 / 𝑥] ¬ 𝜑 ↔ ¬ [𝑦 / 𝑥]𝜑)
42, 3sylib 210 . . . 4 (∀𝑥 ¬ 𝜑 → ¬ [𝑦 / 𝑥]𝜑)
5 df-clab 2753 . . . 4 (𝑦 ∈ {𝑥𝜑} ↔ [𝑦 / 𝑥]𝜑)
64, 5sylnibr 321 . . 3 (∀𝑥 ¬ 𝜑 → ¬ 𝑦 ∈ {𝑥𝜑})
71, 6alrimih 1786 . 2 (∀𝑥 ¬ 𝜑 → ∀𝑦 ¬ 𝑦 ∈ {𝑥𝜑})
8 eq0 4188 . 2 ({𝑥𝜑} = ∅ ↔ ∀𝑦 ¬ 𝑦 ∈ {𝑥𝜑})
97, 8sylibr 226 1 (∀𝑥 ¬ 𝜑 → {𝑥𝜑} = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1505   = wceq 1507  [wsb 2015  wcel 2050  {cab 2752  c0 4172
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-ext 2744
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-clab 2753  df-cleq 2765  df-clel 2840  df-nfc 2912  df-dif 3826  df-nul 4173
This theorem is referenced by:  bj-abf  33717  bj-csbprc  33718
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