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Theorem rab0 3214
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 3196 . . . . 5 ¬ x
21intnanr 821 . . . 4 ¬ (x φ)
3 equid 1562 . . . . 5 x = x
43notnoti 558 . . . 4 ¬ ¬ x = x
52, 42false 601 . . 3 ((x φ) ↔ ¬ x = x)
65abbii 2126 . 2 {x ∣ (x φ)} = {x ∣ ¬ x = x}
7 df-rab 2284 . 2 {x ∅ ∣ φ} = {x ∣ (x φ)}
8 dfnul2 3194 . 2 ∅ = {x ∣ ¬ x = x}
96, 7, 83eqtr4i 2043 1 {x ∅ ∣ φ} = ∅
Colors of variables: wff set class
Syntax hints:  ¬ wn 3   wa 97   = wceq 1223   wcel 1366  {cab 1999  {crab 2279  c0 3192
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 529  ax-in2 530  ax-io 614  ax-5 1309  ax-7 1310  ax-gen 1311  ax-ie1 1355  ax-ie2 1356  ax-8 1368  ax-10 1369  ax-11 1370  ax-i12 1371  ax-bnd 1372  ax-4 1373  ax-17 1392  ax-i9 1396  ax-ial 1400  ax-i5r 1401  ax-ext 1995
This theorem depends on definitions:  df-bi 110  df-tru 1226  df-nf 1323  df-sb 1619  df-clab 2000  df-cleq 2006  df-clel 2009  df-nfc 2140  df-rab 2284  df-v 2528  df-dif 2888  df-nul 3193
This theorem is referenced by: (None)
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