Theorem rab0 4348

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.) (Proof shortened by Umit Teoman Dogan, 10-Jun-2026.)

Assertion

Ref	Expression
rab0	⊢ {𝑥 ∈ ∅ ∣ 𝜑} = ∅

Proof of Theorem rab0

Step	Hyp	Ref	Expression
1		rex0 4322	. . . 4 ⊢ ¬ ∃𝑥 ∈ ∅ ¬ 𝜑
2		dfral2 3122	. . . 4 ⊢ (∀𝑥 ∈ ∅ 𝜑 ↔ ¬ ∃𝑥 ∈ ∅ ¬ 𝜑)
3	1, 2	mpbir 234	. . 3 ⊢ ∀𝑥 ∈ ∅ 𝜑
4	3	rspec 3262	. 2 ⊢ (𝑥 ∈ ∅ → 𝜑)
5	4	rabeqc 3435	1 ⊢ {𝑥 ∈ ∅ ∣ 𝜑} = ∅

Syntax hints:  ¬ wn 3   = wceq 1567  ∀wral 3085  ∃wrex 3095  {crab 3423  ∅c0 4294

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-12 2219  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ral 3086  df-rex 3096  df-rab 3424  df-dif 3916  df-nul 4295

This theorem is referenced by:  rabsnif  4691  fvmptrabfv  7020  supp0  8157  sup00  9421  scott0  9856  psgnfval  19566  pmtrsn  19585  rrgval  20778  00lsp  21076  leftval  28004  rightval  28005  uvtx0  29681  vtxdg0e  29761  wwlksn  30123  wspthsn  30134  iswwlksnon  30139  iswspthsnon  30142  clwwlk0on0  30380  fxpgaval  33424  zar0ring  34209  wevgblacfn  35490  satf0  35759  fvmptrab  47913  fvmptrabdm  47914  prprspr2  48151  initopropdlem  49898  termopropdlem  49899