Theorem abf 4394

Description: A class abstraction determined by a false formula is empty. (Contributed by NM, 20-Jan-2012.) Avoid ax-8 2100, ax-10 2129, ax-11 2146, ax-12 2163. (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 1549	. . 3 ⊢ (𝜑 ↔ ⊥)
3	2	abbii 2794	. 2 ⊢ {𝑥 ∣ 𝜑} = {𝑥 ∣ ⊥}
4		dfnul4 4316	. 2 ⊢ ∅ = {𝑥 ∣ ⊥}
5	3, 4	eqtr4i 2755	1 ⊢ {𝑥 ∣ 𝜑} = ∅

Syntax hints:  ¬ wn 3   = wceq 1533  ⊥wfal 1545  {cab 2701  ∅c0 4314

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-9 2108  ax-ext 2695

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-dif 3943  df-nul 4315

This theorem is referenced by:  csbprc  4398  mpo0  7486  fi0  9411  join0  18360  meet0  18361  addsrid  27797  muls01  27928  mulsrid  27929  n0scut  28119  fmla0disjsuc  34878  0qs  37729  pmapglb2xN  39133