Theorem abf 4358

Description: A class abstraction determined by a false formula is empty. (Contributed by NM, 20-Jan-2012.) Avoid ax-8 2115, ax-10 2146, ax-11 2162, ax-12 2184. (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 2803	. 2 ⊢ {𝑥 ∣ 𝜑} = {𝑥 ∣ ⊥}
4		dfnul4 4287	. 2 ⊢ ∅ = {𝑥 ∣ ⊥}
5	3, 4	eqtr4i 2762	1 ⊢ {𝑥 ∣ 𝜑} = ∅

Syntax hints:  ¬ wn 3   = wceq 1541  ⊥wfal 1553  {cab 2714  ∅c0 4285

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 2123  ax-ext 2708

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 2715  df-cleq 2728  df-dif 3904  df-nul 4286

This theorem is referenced by:  csbprc  4361  mpo0  7443  0qs  8700  fi0  9323  join0  18326  meet0  18327  addsrid  27960  muls01  28108  mulsrid  28109  onaddscl  28273  onmulscl  28274  n0cut  28330  fmla0disjsuc  35592  pmapglb2xN  40032