Theorem abf 4363

Description: A class abstraction determined by a false formula is empty. (Contributed by NM, 20-Jan-2012.) Avoid ax-8 2147, ax-10 2178, ax-11 2194, ax-12 2215. (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 1579	. . 3 ⊢ (𝜑 ↔ ⊥)
3	2	abbii 2832	. 2 ⊢ {𝑥 ∣ 𝜑} = {𝑥 ∣ ⊥}
4		dfnul4 4290	. 2 ⊢ ∅ = {𝑥 ∣ ⊥}
5	3, 4	eqtr4i 2791	1 ⊢ {𝑥 ∣ 𝜑} = ∅

Syntax hints:  ¬ wn 3   = wceq 1563  ⊥wfal 1575  {cab 2743  ∅c0 4288

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-9 2155  ax-ext 2737

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-dif 3910  df-nul 4289

This theorem is referenced by:  mpo0  7485  0qs  8748  fi0  9368  join0  18447  meet0  18448  addsrid  28111  muls01  28259  mulsrid  28260  onaddscl  28424  onmulscl  28425  n0cut  28481  fmla0disjsuc  35756  pmapglb2xN  40403