Theorem abf 4398

Description: A class abstraction determined by a false formula is empty. (Contributed by NM, 20-Jan-2012.) Avoid ax-8 2101, ax-10 2130, ax-11 2147, ax-12 2164. (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 1550	. . 3 ⊢ (𝜑 ↔ ⊥)
3	2	abbii 2797	. 2 ⊢ {𝑥 ∣ 𝜑} = {𝑥 ∣ ⊥}
4		dfnul4 4320	. 2 ⊢ ∅ = {𝑥 ∣ ⊥}
5	3, 4	eqtr4i 2758	1 ⊢ {𝑥 ∣ 𝜑} = ∅

Syntax hints:  ¬ wn 3   = wceq 1534  ⊥wfal 1546  {cab 2704  ∅c0 4318

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-9 2109  ax-ext 2698

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2705  df-cleq 2719  df-dif 3947  df-nul 4319

This theorem is referenced by:  csbprc  4402  mpo0  7499  fi0  9435  join0  18388  meet0  18389  addsrid  27868  muls01  27999  mulsrid  28000  n0scut  28190  fmla0disjsuc  34944  0qs  37778  pmapglb2xN  39182