Theorem abf 4364

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 2213. (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 2828	. 2 ⊢ {𝑥 ∣ 𝜑} = {𝑥 ∣ ⊥}
4		dfnul4 4281	. 2 ⊢ ∅ = {𝑥 ∣ ⊥}
5	3, 4	eqtr4i 2787	1 ⊢ {𝑥 ∣ 𝜑} = ∅

This proof depends on syntax axioms:  ¬ wn 3   = wceq 1570  ⊥wfal 1582  {cab 2739  ∅c0 4279

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 2155  ax-ext 2733

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 2740  df-cleq 2753  df-dif 3902  df-nul 4280

This theorem is used by:  mpo0  7497  0qs  8767  fi0  9396  join0  18557  meet0  18558  addsrid  28332  muls01  28480  mulsrid  28481  onaddscl  28645  onmulscl  28646  n0cut  28702  fmla0disjsuc  36132  pmapglb2xN  40797