MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  rab0 Structured version   Visualization version   GIF version

Theorem rab0 4392
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 3434 . 2 {𝑥 ∈ ∅ ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ ∅ ∧ 𝜑)}
2 ab0 4386 . . 3 ({𝑥 ∣ (𝑥 ∈ ∅ ∧ 𝜑)} = ∅ ↔ ∀𝑥 ¬ (𝑥 ∈ ∅ ∧ 𝜑))
3 noel 4344 . . . 4 ¬ 𝑥 ∈ ∅
43intnanr 487 . . 3 ¬ (𝑥 ∈ ∅ ∧ 𝜑)
52, 4mpgbir 1796 . 2 {𝑥 ∣ (𝑥 ∈ ∅ ∧ 𝜑)} = ∅
61, 5eqtri 2763 1 {𝑥 ∈ ∅ ∣ 𝜑} = ∅
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 395   = wceq 1537  wcel 2106  {cab 2712  {crab 3433  c0 4339
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-dif 3966  df-nul 4340
This theorem is referenced by:  rabsnif  4728  fvmptrabfv  7048  supp0  8189  sup00  9502  scott0  9924  psgnfval  19533  pmtrsn  19552  rrgval  20714  00lsp  20997  leftval  27917  rightval  27918  uvtx0  29426  vtxdg0e  29507  wwlksn  29867  wspthsn  29878  iswwlksnon  29883  iswspthsnon  29886  clwwlk0on0  30121  zar0ring  33839  wevgblacfn  35093  satf0  35357  fvmptrab  47242  fvmptrabdm  47243  prprspr2  47443
  Copyright terms: Public domain W3C validator