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Theorem bj-ab0 33225
Description: The class of sets verifying a falsity is the empty set (closed form of abf 4120). (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 1990 . . 3 (∀𝑥 ¬ 𝜑 → ∀𝑦𝑥 ¬ 𝜑)
2 bj-stdpc4v 33084 . . . . 5 (∀𝑥 ¬ 𝜑 → [𝑦 / 𝑥] ¬ 𝜑)
3 sbn 2537 . . . . 5 ([𝑦 / 𝑥] ¬ 𝜑 ↔ ¬ [𝑦 / 𝑥]𝜑)
42, 3sylib 208 . . . 4 (∀𝑥 ¬ 𝜑 → ¬ [𝑦 / 𝑥]𝜑)
5 df-clab 2757 . . . 4 (𝑦 ∈ {𝑥𝜑} ↔ [𝑦 / 𝑥]𝜑)
64, 5sylnibr 318 . . 3 (∀𝑥 ¬ 𝜑 → ¬ 𝑦 ∈ {𝑥𝜑})
71, 6alrimih 1898 . 2 (∀𝑥 ¬ 𝜑 → ∀𝑦 ¬ 𝑦 ∈ {𝑥𝜑})
8 eq0 4074 . 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 1628   = wceq 1630  [wsb 2048  wcel 2144  {cab 2756  c0 4061
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-v 3351  df-dif 3724  df-nul 4062
This theorem is referenced by:  bj-abf  33226  bj-csbprc  33227
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