Theorem rab0 4333

Description: Any restricted class abstraction restricted to the empty set is empty. (Contributed by NM, 15-Oct-2003.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) (Proof shortened by JJ, 14-Jul-2021.)

Assertion

Ref	Expression
rab0	⊢ {𝑥 ∈ ∅ ∣ 𝜑} = ∅

Proof of Theorem rab0

Step	Hyp	Ref	Expression
1		df-rab 3396	. 2 ⊢ {𝑥 ∈ ∅ ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ ∅ ∧ 𝜑)}
2		ab0 4327	. . 3 ⊢ ({𝑥 ∣ (𝑥 ∈ ∅ ∧ 𝜑)} = ∅ ↔ ∀𝑥 ¬ (𝑥 ∈ ∅ ∧ 𝜑))
3		noel 4285	. . . 4 ⊢ ¬ 𝑥 ∈ ∅
4	3	intnanr 487	. . 3 ⊢ ¬ (𝑥 ∈ ∅ ∧ 𝜑)
5	2, 4	mpgbir 1800	. 2 ⊢ {𝑥 ∣ (𝑥 ∈ ∅ ∧ 𝜑)} = ∅
6	1, 5	eqtri 2754	1 ⊢ {𝑥 ∈ ∅ ∣ 𝜑} = ∅

Syntax hints:  ¬ wn 3   ∧ wa 395   = wceq 1541   ∈ wcel 2111  {cab 2709  {crab 3395  ∅c0 4280

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-rab 3396  df-dif 3900  df-nul 4281

This theorem is referenced by:  rabsnif  4673  fvmptrabfv  6961  supp0  8095  sup00  9349  scott0  9779  psgnfval  19412  pmtrsn  19431  rrgval  20612  00lsp  20914  leftval  27804  rightval  27805  uvtx0  29372  vtxdg0e  29453  wwlksn  29815  wspthsn  29826  iswwlksnon  29831  iswspthsnon  29834  clwwlk0on0  30072  fxpgaval  33136  zar0ring  33891  wevgblacfn  35153  satf0  35416  fvmptrab  47331  fvmptrabdm  47332  prprspr2  47557  initopropdlem  49280  termopropdlem  49281