Theorem abf 4412

Description: A class abstraction determined by a false formula is empty. (Contributed by NM, 20-Jan-2012.) Avoid ax-8 2108, ax-10 2139, ax-11 2155, ax-12 2175. (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 2807	. 2 ⊢ {𝑥 ∣ 𝜑} = {𝑥 ∣ ⊥}
4		dfnul4 4341	. 2 ⊢ ∅ = {𝑥 ∣ ⊥}
5	3, 4	eqtr4i 2766	1 ⊢ {𝑥 ∣ 𝜑} = ∅

Syntax hints:  ¬ wn 3   = wceq 1537  ⊥wfal 1549  {cab 2712  ∅c0 4339

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-9 2116  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-dif 3966  df-nul 4340

This theorem is referenced by:  csbprc  4415  mpo0  7518  0qs  8806  fi0  9458  join0  18463  meet0  18464  addsrid  28012  muls01  28153  mulsrid  28154  onaddscl  28301  onmulscl  28302  n0scut  28353  1p1e2s  28415  fmla0disjsuc  35383  pmapglb2xN  39755