Theorem abf 4403

Description: A class abstraction determined by a false formula is empty. (Contributed by NM, 20-Jan-2012.) Avoid ax-8 2109, ax-10 2138, ax-11 2155, ax-12 2172. (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	1	bifal 1558	. . 3 ⊢ (𝜑 ↔ ⊥)
3	2	abbii 2803	. 2 ⊢ {𝑥 ∣ 𝜑} = {𝑥 ∣ ⊥}
4		dfnul4 4325	. 2 ⊢ ∅ = {𝑥 ∣ ⊥}
5	3, 4	eqtr4i 2764	1 ⊢ {𝑥 ∣ 𝜑} = ∅

Syntax hints:  ¬ wn 3   = wceq 1542  ⊥wfal 1554  {cab 2710  ∅c0 4323

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-9 2117  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-dif 3952  df-nul 4324

This theorem is referenced by:  csbprc  4407  mpo0  7494  fi0  9415  join0  18358  meet0  18359  addsrid  27448  muls01  27568  mulsrid  27569  fmla0disjsuc  34389  0qs  37239  pmapglb2xN  38643