Theorem abf 4360

Description: A class abstraction determined by a false formula is empty. (Contributed by NM, 20-Jan-2012.) Avoid ax-8 2116, ax-10 2147, ax-11 2163, ax-12 2185. (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 1558	. . 3 ⊢ (𝜑 ↔ ⊥)
3	2	abbii 2804	. 2 ⊢ {𝑥 ∣ 𝜑} = {𝑥 ∣ ⊥}
4		dfnul4 4289	. 2 ⊢ ∅ = {𝑥 ∣ ⊥}
5	3, 4	eqtr4i 2763	1 ⊢ {𝑥 ∣ 𝜑} = ∅

Syntax hints:  ¬ wn 3   = wceq 1542  ⊥wfal 1554  {cab 2715  ∅c0 4287

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 1912  ax-6 1969  ax-7 2010  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-dif 3906  df-nul 4288

This theorem is referenced by:  csbprc  4363  mpo0  7453  0qs  8711  fi0  9335  join0  18338  meet0  18339  addsrid  27972  muls01  28120  mulsrid  28121  onaddscl  28285  onmulscl  28286  n0cut  28342  fmla0disjsuc  35611  pmapglb2xN  40145