Theorem rab0 4338

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 4311	. . . 4 ⊢ ¬ ∃𝑥 ∈ ∅ ¬ 𝜑
2		dfral2 3117	. . . 4 ⊢ (∀𝑥 ∈ ∅ 𝜑 ↔ ¬ ∃𝑥 ∈ ∅ ¬ 𝜑)
3	1, 2	mpbir 234	. . 3 ⊢ ∀𝑥 ∈ ∅ 𝜑
4	3	rspec 3257	. 2 ⊢ (𝑥 ∈ ∅ → 𝜑)
5	4	rabeqc 3428	1 ⊢ {𝑥 ∈ ∅ ∣ 𝜑} = ∅

This proof depends on syntax axioms:  ¬ wn 3   = wceq 1570  ∀wral 3080  ∃wrex 3090  {crab 3416  ∅c0 4281

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-12 2216  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-rab 3417  df-dif 4050  df-nul 4282

This theorem is used by:  rabsnif  4686  fvmptrabfv  7057  supp0  8220  sup00  9494  scott0OLD  9975  psgnfval  19765  pmtrsn  19784  rrgval  21000  00lsp  21307  leftval  28292  rightval  28293  uvtx0  30032  vtxdg0e  30112  wwlksn  30483  wspthsn  30494  iswwlksnon  30499  iswspthsnon  30502  clwwlk0on0  30740  fxpgaval  33827  zar0ring  34621  wevgblacfn  36080  satf0  36345  fvmptrab  48635  fvmptrabdm  48636  prprspr2  48873  initopropdlem  50693  termopropdlem  50694