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Theorem rab0 3297
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 3276 . . . . 5 ¬ 𝑥 ∈ ∅
21intnanr 875 . . . 4 ¬ (𝑥 ∈ ∅ ∧ 𝜑)
3 equid 1632 . . . . 5 𝑥 = 𝑥
43notnoti 607 . . . 4 ¬ ¬ 𝑥 = 𝑥
52, 42false 650 . . 3 ((𝑥 ∈ ∅ ∧ 𝜑) ↔ ¬ 𝑥 = 𝑥)
65abbii 2200 . 2 {𝑥 ∣ (𝑥 ∈ ∅ ∧ 𝜑)} = {𝑥 ∣ ¬ 𝑥 = 𝑥}
7 df-rab 2364 . 2 {𝑥 ∈ ∅ ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ ∅ ∧ 𝜑)}
8 dfnul2 3274 . 2 ∅ = {𝑥 ∣ ¬ 𝑥 = 𝑥}
96, 7, 83eqtr4i 2115 1 {𝑥 ∈ ∅ ∣ 𝜑} = ∅
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wa 102   = wceq 1287  wcel 1436  {cab 2071  {crab 2359  c0 3272
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-rab 2364  df-v 2616  df-dif 2988  df-nul 3273
This theorem is referenced by:  ssfirab  6571  sup00  6619
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