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Theorem rab0 3240
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 3222 . . . . 5 ¬ x
21intnanr 838 . . . 4 ¬ (x φ)
3 equid 1586 . . . . 5 x = x
43notnoti 573 . . . 4 ¬ ¬ x = x
52, 42false 616 . . 3 ((x φ) ↔ ¬ x = x)
65abbii 2150 . 2 {x ∣ (x φ)} = {x ∣ ¬ x = x}
7 df-rab 2309 . 2 {x ∅ ∣ φ} = {x ∣ (x φ)}
8 dfnul2 3220 . 2 ∅ = {x ∣ ¬ x = x}
96, 7, 83eqtr4i 2067 1 {x ∅ ∣ φ} = ∅
Colors of variables: wff set class
Syntax hints:  ¬ wn 3   wa 97   = wceq 1242   wcel 1390  {cab 2023  {crab 2304  c0 3218
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-rab 2309  df-v 2553  df-dif 2914  df-nul 3219
This theorem is referenced by: (None)
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