Theorem abf 4429

Description: A class abstraction determined by a false formula is empty. (Contributed by NM, 20-Jan-2012.) Avoid ax-8 2110, ax-10 2141, ax-11 2158, ax-12 2178. (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 1553	. . 3 ⊢ (𝜑 ↔ ⊥)
3	2	abbii 2812	. 2 ⊢ {𝑥 ∣ 𝜑} = {𝑥 ∣ ⊥}
4		dfnul4 4354	. 2 ⊢ ∅ = {𝑥 ∣ ⊥}
5	3, 4	eqtr4i 2771	1 ⊢ {𝑥 ∣ 𝜑} = ∅

Syntax hints:  ¬ wn 3   = wceq 1537  ⊥wfal 1549  {cab 2717  ∅c0 4352

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-dif 3979  df-nul 4353

This theorem is referenced by:  csbprc  4432  mpo0  7535  0qs  8825  fi0  9489  join0  18475  meet0  18476  addsrid  28015  muls01  28156  mulsrid  28157  onaddscl  28304  onmulscl  28305  n0scut  28356  1p1e2s  28418  fmla0disjsuc  35366  pmapglb2xN  39729