Theorem abf 4363

Description: A class abstraction determined by a false formula is empty. (Contributed by NM, 20-Jan-2012.) Avoid ax-8 2146, ax-10 2177, ax-11 2193, ax-12 2212. (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 1585	. . 3 ⊢ (𝜑 ↔ ⊥)
3	2	abbii 2827	. 2 ⊢ {𝑥 ∣ 𝜑} = {𝑥 ∣ ⊥}
4		dfnul4 4280	. 2 ⊢ ∅ = {𝑥 ∣ ⊥}
5	3, 4	eqtr4i 2786	1 ⊢ {𝑥 ∣ 𝜑} = ∅

This proof depends on syntax axioms:  ¬ wn 3   = wceq 1569  ⊥wfal 1581  {cab 2738  ∅c0 4278

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-dif 4047  df-nul 4279

This theorem is used by:  mpo0  7505  0qs  8786  fi0  9415  join0  18582  meet0  18583  addsrid  28355  muls01  28503  mulsrid  28504  onaddscl  28668  onmulscl  28669  n0cut  28725  fmla0disjsuc  36249  pmapglb2xN  40922