Theorem abf 4386

Description: A class abstraction determined by a false formula is empty. (Contributed by NM, 20-Jan-2012.) Avoid ax-8 2111, ax-10 2142, 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 1556	. . 3 ⊢ (𝜑 ↔ ⊥)
3	2	abbii 2803	. 2 ⊢ {𝑥 ∣ 𝜑} = {𝑥 ∣ ⊥}
4		dfnul4 4315	. 2 ⊢ ∅ = {𝑥 ∣ ⊥}
5	3, 4	eqtr4i 2762	1 ⊢ {𝑥 ∣ 𝜑} = ∅

Syntax hints:  ¬ wn 3   = wceq 1540  ⊥wfal 1552  {cab 2714  ∅c0 4313

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-9 2119  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2715  df-cleq 2728  df-dif 3934  df-nul 4314

This theorem is referenced by:  csbprc  4389  mpo0  7497  0qs  8786  fi0  9437  join0  18420  meet0  18421  addsrid  27928  muls01  28072  mulsrid  28073  onaddscl  28231  onmulscl  28232  n0scut  28283  1p1e2s  28359  fmla0disjsuc  35425  pmapglb2xN  39796