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Theorem abf 4363
Description: A class abstraction determined by a false formula is empty. (Contributed by NM, 20-Jan-2012.) Avoid ax-8 2109, ax-10 2138, ax-11 2155, ax-12 2172. (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
21bifal 1558 . . 3 (𝜑 ↔ ⊥)
32abbii 2803 . 2 {𝑥𝜑} = {𝑥 ∣ ⊥}
4 dfnul4 4285 . 2 ∅ = {𝑥 ∣ ⊥}
53, 4eqtr4i 2764 1 {𝑥𝜑} = ∅
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1542  wfal 1554  {cab 2710  c0 4283
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-dif 3914  df-nul 4284
This theorem is referenced by:  csbprc  4367  mpo0  7443  fi0  9361  join0  18299  meet0  18300  addsrid  27298  muls01  27397  muls02  27398  mulsrid  27399  mulslid  27400  fmla0disjsuc  34049  0qs  36877  pmapglb2xN  38281
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