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Theorem rab0 3361
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.)
Assertion
Ref Expression
rab0 {𝑥 ∈ ∅ ∣ 𝜑} = ∅

Proof of Theorem rab0
StepHypRef Expression
1 noel 3337 . . . . 5 ¬ 𝑥 ∈ ∅
21intnanr 900 . . . 4 ¬ (𝑥 ∈ ∅ ∧ 𝜑)
3 equid 1662 . . . . 5 𝑥 = 𝑥
43notnoti 619 . . . 4 ¬ ¬ 𝑥 = 𝑥
52, 42false 675 . . 3 ((𝑥 ∈ ∅ ∧ 𝜑) ↔ ¬ 𝑥 = 𝑥)
65abbii 2233 . 2 {𝑥 ∣ (𝑥 ∈ ∅ ∧ 𝜑)} = {𝑥 ∣ ¬ 𝑥 = 𝑥}
7 df-rab 2402 . 2 {𝑥 ∈ ∅ ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ ∅ ∧ 𝜑)}
8 dfnul2 3335 . 2 ∅ = {𝑥 ∣ ¬ 𝑥 = 𝑥}
96, 7, 83eqtr4i 2148 1 {𝑥 ∈ ∅ ∣ 𝜑} = ∅
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wa 103   = wceq 1316  wcel 1465  {cab 2103  {crab 2397  c0 3333
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-rab 2402  df-v 2662  df-dif 3043  df-nul 3334
This theorem is referenced by:  ssfirab  6790  sup00  6858
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