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Theorem bj-ab0 33129
Description: The class of sets verifying a falsity is the empty set (closed form of abf 4086). (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 1952 . . 3 (∀𝑥 ¬ 𝜑 → ∀𝑦𝑥 ¬ 𝜑)
2 bj-stdpc4v 32981 . . . . 5 (∀𝑥 ¬ 𝜑 → [𝑦 / 𝑥] ¬ 𝜑)
3 sbn 2492 . . . . 5 ([𝑦 / 𝑥] ¬ 𝜑 ↔ ¬ [𝑦 / 𝑥]𝜑)
42, 3sylib 208 . . . 4 (∀𝑥 ¬ 𝜑 → ¬ [𝑦 / 𝑥]𝜑)
5 df-clab 2711 . . . 4 (𝑦 ∈ {𝑥𝜑} ↔ [𝑦 / 𝑥]𝜑)
64, 5sylnibr 318 . . 3 (∀𝑥 ¬ 𝜑 → ¬ 𝑦 ∈ {𝑥𝜑})
71, 6alrimih 1864 . 2 (∀𝑥 ¬ 𝜑 → ∀𝑦 ¬ 𝑦 ∈ {𝑥𝜑})
8 eq0 4037 . 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 1594   = wceq 1596  [wsb 2010  wcel 2103  {cab 2710  c0 4023
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-v 3306  df-dif 3683  df-nul 4024
This theorem is referenced by:  bj-abf  33130  bj-csbprc  33131
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