Theorem abf 4310

Description: A class abstraction determined by a false formula is empty. (Contributed by NM, 20-Jan-2012.) Avoid ax-8 2113, ax-10 2142, ax-11 2158, ax-12 2175. (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		equid 2019	. . . . 5 ⊢ 𝑥 = 𝑥
3	2	notnoti 145	. . . 4 ⊢ ¬ ¬ 𝑥 = 𝑥
4	1, 3	2false 379	. . 3 ⊢ (𝜑 ↔ ¬ 𝑥 = 𝑥)
5	4	abbii 2863	. 2 ⊢ {𝑥 ∣ 𝜑} = {𝑥 ∣ ¬ 𝑥 = 𝑥}
6		dfnul2 4245	. 2 ⊢ ∅ = {𝑥 ∣ ¬ 𝑥 = 𝑥}
7	5, 6	eqtr4i 2824	1 ⊢ {𝑥 ∣ 𝜑} = ∅

Syntax hints:  ¬ wn 3   = wceq 1538  {cab 2776  ∅c0 4243

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-9 2121  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-dif 3884  df-nul 4244

This theorem is referenced by:  csbprc  4313  mpo0  7218  fi0  8868  meet0  17739  join0  17740  fmla0disjsuc  32758  0qs  35782  pmapglb2xN  37068