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Theorem rab0 3357
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 3333 . . . . 5 ¬ 𝑥 ∈ ∅
21intnanr 898 . . . 4 ¬ (𝑥 ∈ ∅ ∧ 𝜑)
3 equid 1660 . . . . 5 𝑥 = 𝑥
43notnoti 617 . . . 4 ¬ ¬ 𝑥 = 𝑥
52, 42false 673 . . 3 ((𝑥 ∈ ∅ ∧ 𝜑) ↔ ¬ 𝑥 = 𝑥)
65abbii 2230 . 2 {𝑥 ∣ (𝑥 ∈ ∅ ∧ 𝜑)} = {𝑥 ∣ ¬ 𝑥 = 𝑥}
7 df-rab 2399 . 2 {𝑥 ∈ ∅ ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ ∅ ∧ 𝜑)}
8 dfnul2 3331 . 2 ∅ = {𝑥 ∣ ¬ 𝑥 = 𝑥}
96, 7, 83eqtr4i 2145 1 {𝑥 ∈ ∅ ∣ 𝜑} = ∅
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wa 103   = wceq 1314  wcel 1463  {cab 2101  {crab 2394  c0 3329
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-rab 2399  df-v 2659  df-dif 3039  df-nul 3330
This theorem is referenced by:  ssfirab  6774  sup00  6842
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