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Theorem bj-ab0 34830
Description: The class of sets verifying a falsity is the empty set (closed form of abf 4317). (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 2074 . . . 4 (∀𝑥 ¬ 𝜑 → [𝑦 / 𝑥] ¬ 𝜑)
2 sbn1 2109 . . . 4 ([𝑦 / 𝑥] ¬ 𝜑 → ¬ [𝑦 / 𝑥]𝜑)
31, 2syl 17 . . 3 (∀𝑥 ¬ 𝜑 → ¬ [𝑦 / 𝑥]𝜑)
4 df-clab 2715 . . 3 (𝑦 ∈ {𝑥𝜑} ↔ [𝑦 / 𝑥]𝜑)
53, 4sylnibr 332 . 2 (∀𝑥 ¬ 𝜑 → ¬ 𝑦 ∈ {𝑥𝜑})
65eq0rdv 4319 1 (∀𝑥 ¬ 𝜑 → {𝑥𝜑} = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1541   = wceq 1543  [wsb 2070  wcel 2110  {cab 2714  c0 4237
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-9 2120  ax-ext 2708
This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2071  df-clab 2715  df-cleq 2729  df-dif 3869  df-nul 4238
This theorem is referenced by:  bj-abf  34831  bj-csbprc  34832
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