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Theorem abf 4310
 Description: A class abstraction determined by a false formula is empty. (Contributed by NM, 20-Jan-2012.) Avoid ax-8 2113, ax-10 2142, ax-11 2158, ax-12 2175. (Revised by Gino Giotto, 30-Jun-2024.)
Hypothesis
Ref Expression
abf.1 ¬ 𝜑
Assertion
Ref Expression
abf {𝑥𝜑} = ∅

Proof of Theorem abf
StepHypRef Expression
1 abf.1 . . . 4 ¬ 𝜑
2 equid 2019 . . . . 5 𝑥 = 𝑥
32notnoti 145 . . . 4 ¬ ¬ 𝑥 = 𝑥
41, 32false 379 . . 3 (𝜑 ↔ ¬ 𝑥 = 𝑥)
54abbii 2863 . 2 {𝑥𝜑} = {𝑥 ∣ ¬ 𝑥 = 𝑥}
6 dfnul2 4245 . 2 ∅ = {𝑥 ∣ ¬ 𝑥 = 𝑥}
75, 6eqtr4i 2824 1 {𝑥𝜑} = ∅
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   = wceq 1538  {cab 2776  ∅c0 4243 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-9 2121  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-dif 3884  df-nul 4244 This theorem is referenced by:  csbprc  4313  mpo0  7218  fi0  8870  meet0  17741  join0  17742  fmla0disjsuc  32770  0qs  35798  pmapglb2xN  37084
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