Theorem abf 4405

Description: A class abstraction determined by a false formula is empty. (Contributed by NM, 20-Jan-2012.) Avoid ax-8 2109, ax-10 2140, ax-11 2156, ax-12 2176. (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 1555	. . 3 ⊢ (𝜑 ↔ ⊥)
3	2	abbii 2808	. 2 ⊢ {𝑥 ∣ 𝜑} = {𝑥 ∣ ⊥}
4		dfnul4 4334	. 2 ⊢ ∅ = {𝑥 ∣ ⊥}
5	3, 4	eqtr4i 2767	1 ⊢ {𝑥 ∣ 𝜑} = ∅

Syntax hints:  ¬ wn 3   = wceq 1539  ⊥wfal 1551  {cab 2713  ∅c0 4332

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-9 2117  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-dif 3953  df-nul 4333

This theorem is referenced by:  csbprc  4408  mpo0  7519  0qs  8808  fi0  9461  join0  18451  meet0  18452  addsrid  27998  muls01  28139  mulsrid  28140  onaddscl  28287  onmulscl  28288  n0scut  28339  1p1e2s  28401  fmla0disjsuc  35404  pmapglb2xN  39775