Theorem rab0 4340

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 3402	. 2 ⊢ {𝑥 ∈ ∅ ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ ∅ ∧ 𝜑)}
2		ab0 4334	. . 3 ⊢ ({𝑥 ∣ (𝑥 ∈ ∅ ∧ 𝜑)} = ∅ ↔ ∀𝑥 ¬ (𝑥 ∈ ∅ ∧ 𝜑))
3		noel 4292	. . . 4 ⊢ ¬ 𝑥 ∈ ∅
4	3	intnanr 487	. . 3 ⊢ ¬ (𝑥 ∈ ∅ ∧ 𝜑)
5	2, 4	mpgbir 1801	. 2 ⊢ {𝑥 ∣ (𝑥 ∈ ∅ ∧ 𝜑)} = ∅
6	1, 5	eqtri 2760	1 ⊢ {𝑥 ∈ ∅ ∣ 𝜑} = ∅

Syntax hints:  ¬ wn 3   ∧ wa 395   = wceq 1542   ∈ wcel 2114  {cab 2715  {crab 3401  ∅c0 4287

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-rab 3402  df-dif 3906  df-nul 4288

This theorem is referenced by:  rabsnif  4682  fvmptrabfv  6982  supp0  8117  sup00  9380  scott0  9810  psgnfval  19441  pmtrsn  19460  rrgval  20642  00lsp  20944  leftval  27857  rightval  27858  uvtx0  29479  vtxdg0e  29560  wwlksn  29922  wspthsn  29933  iswwlksnon  29938  iswspthsnon  29941  clwwlk0on0  30179  fxpgaval  33261  zar0ring  34056  wevgblacfn  35325  satf0  35588  fvmptrab  47652  fvmptrabdm  47653  prprspr2  47878  initopropdlem  49599  termopropdlem  49600