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Theorem rab0 3929
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
StepHypRef Expression
1 df-rab 2916 . 2 {𝑥 ∈ ∅ ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ ∅ ∧ 𝜑)}
2 ab0 3925 . . 3 ({𝑥 ∣ (𝑥 ∈ ∅ ∧ 𝜑)} = ∅ ↔ ∀𝑥 ¬ (𝑥 ∈ ∅ ∧ 𝜑))
3 noel 3895 . . . 4 ¬ 𝑥 ∈ ∅
43intnanr 960 . . 3 ¬ (𝑥 ∈ ∅ ∧ 𝜑)
52, 4mpgbir 1723 . 2 {𝑥 ∣ (𝑥 ∈ ∅ ∧ 𝜑)} = ∅
61, 5eqtri 2643 1 {𝑥 ∈ ∅ ∣ 𝜑} = ∅
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 384   = wceq 1480  wcel 1987  {cab 2607  {crab 2911  c0 3891
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-rab 2916  df-v 3188  df-dif 3558  df-nul 3892
This theorem is referenced by:  rabsnif  4228  supp0  7245  sup00  8314  scott0  8693  psgnfval  17841  pmtrsn  17860  00lsp  18900  rrgval  19206  uvtxa0  26181  vtxdg0e  26256  wwlksn  26598  wspthsn  26604  iswwlksnon  26609  iswspthsnon  26610  clwwlksn  26748
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