Theorem abf 4374

Description: A class abstraction determined by a false formula is empty. (Contributed by NM, 20-Jan-2012.) Avoid ax-8 2148, ax-10 2179, ax-11 2195, ax-12 2216. (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 1586	. . 3 ⊢ (𝜑 ↔ ⊥)
3	2	abbii 2833	. 2 ⊢ {𝑥 ∣ 𝜑} = {𝑥 ∣ ⊥}
4		dfnul4 4291	. 2 ⊢ ∅ = {𝑥 ∣ ⊥}
5	3, 4	eqtr4i 2792	1 ⊢ {𝑥 ∣ 𝜑} = ∅

This proof depends on syntax axioms:  ¬ wn 3   = wceq 1570  ⊥wfal 1582  {cab 2744  ∅c0 4289

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-9 2156  ax-ext 2738

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2745  df-cleq 2758  df-dif 3911  df-nul 4290

This theorem is used by:  mpo0  7508  0qs  8769  fi0  9390  join0  18484  meet0  18485  addsrid  28194  muls01  28342  mulsrid  28343  onaddscl  28507  onmulscl  28508  n0cut  28564  fmla0disjsuc  35911  pmapglb2xN  40587