Theorem rab0 4337

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 3395	. 2 ⊢ {𝑥 ∈ ∅ ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ ∅ ∧ 𝜑)}
2		ab0 4331	. . 3 ⊢ ({𝑥 ∣ (𝑥 ∈ ∅ ∧ 𝜑)} = ∅ ↔ ∀𝑥 ¬ (𝑥 ∈ ∅ ∧ 𝜑))
3		noel 4289	. . . 4 ⊢ ¬ 𝑥 ∈ ∅
4	3	intnanr 487	. . 3 ⊢ ¬ (𝑥 ∈ ∅ ∧ 𝜑)
5	2, 4	mpgbir 1799	. 2 ⊢ {𝑥 ∣ (𝑥 ∈ ∅ ∧ 𝜑)} = ∅
6	1, 5	eqtri 2752	1 ⊢ {𝑥 ∈ ∅ ∣ 𝜑} = ∅

Syntax hints:  ¬ wn 3   ∧ wa 395   = wceq 1540   ∈ wcel 2109  {cab 2707  {crab 3394  ∅c0 4284

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-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-rab 3395  df-dif 3906  df-nul 4285

This theorem is referenced by:  rabsnif  4675  fvmptrabfv  6962  supp0  8098  sup00  9355  scott0  9782  psgnfval  19379  pmtrsn  19398  rrgval  20582  00lsp  20884  leftval  27773  rightval  27774  uvtx0  29339  vtxdg0e  29420  wwlksn  29782  wspthsn  29793  iswwlksnon  29798  iswspthsnon  29801  clwwlk0on0  30036  fxpgaval  33109  zar0ring  33845  wevgblacfn  35082  satf0  35345  fvmptrab  47276  fvmptrabdm  47277  prprspr2  47502  initopropdlem  49225  termopropdlem  49226