Theorem abf 4356

Description: A class abstraction determined by a false formula is empty. (Contributed by NM, 20-Jan-2012.) Avoid ax-8 2113, ax-10 2144, ax-11 2160, ax-12 2180. (Revised by GG, 30-Jun-2024.)

Hypothesis

Ref	Expression
abf.1	⊢ ¬ 𝜑

Assertion

Ref	Expression
abf	⊢ {𝑥 ∣ 𝜑} = ∅

Proof of Theorem abf

Step	Hyp	Ref	Expression
1		abf.1	. . . 4 ⊢ ¬ 𝜑
2	1	bifal 1557	. . 3 ⊢ (𝜑 ↔ ⊥)
3	2	abbii 2798	. 2 ⊢ {𝑥 ∣ 𝜑} = {𝑥 ∣ ⊥}
4		dfnul4 4285	. 2 ⊢ ∅ = {𝑥 ∣ ⊥}
5	3, 4	eqtr4i 2757	1 ⊢ {𝑥 ∣ 𝜑} = ∅

Syntax hints:  ¬ wn 3   = wceq 1541  ⊥wfal 1553  {cab 2709  ∅c0 4283

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-dif 3905  df-nul 4284

This theorem is referenced by:  csbprc  4359  mpo0  7431  0qs  8687  fi0  9304  join0  18309  meet0  18310  addsrid  27908  muls01  28052  mulsrid  28053  onaddscl  28211  onmulscl  28212  n0scut  28263  1p1e2s  28340  fmla0disjsuc  35440  pmapglb2xN  39817