Theorem abf 4333

Description: A class abstraction determined by a false formula is empty. (Contributed by NM, 20-Jan-2012.) Avoid ax-8 2110, ax-10 2139, ax-11 2156, ax-12 2173. (Revised by Gino Giotto, 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 1555	. . 3 ⊢ (𝜑 ↔ ⊥)
3	2	abbii 2809	. 2 ⊢ {𝑥 ∣ 𝜑} = {𝑥 ∣ ⊥}
4		dfnul4 4255	. 2 ⊢ ∅ = {𝑥 ∣ ⊥}
5	3, 4	eqtr4i 2769	1 ⊢ {𝑥 ∣ 𝜑} = ∅

Syntax hints:  ¬ wn 3   = wceq 1539  ⊥wfal 1551  {cab 2715  ∅c0 4253

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-dif 3886  df-nul 4254

This theorem is referenced by:  csbprc  4337  mpo0  7338  fi0  9109  join0  18038  meet0  18039  fmla0disjsuc  33260  addsid1  34054  0qs  36427  pmapglb2xN  37713