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Theorem rab0 3224
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 3206 . . . . 5 ¬ x
21intnanr 827 . . . 4 ¬ (x φ)
3 equid 1570 . . . . 5 x = x
43notnoti 561 . . . 4 ¬ ¬ x = x
52, 42false 604 . . 3 ((x φ) ↔ ¬ x = x)
65abbii 2135 . 2 {x ∣ (x φ)} = {x ∣ ¬ x = x}
7 df-rab 2291 . 2 {x ∅ ∣ φ} = {x ∣ (x φ)}
8 dfnul2 3204 . 2 ∅ = {x ∣ ¬ x = x}
96, 7, 83eqtr4i 2052 1 {x ∅ ∣ φ} = ∅
Colors of variables: wff set class
Syntax hints:  ¬ wn 3   wa 97   = wceq 1373   wcel 1375  {cab 2008  {crab 2286  c0 3202
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 532  ax-in2 533  ax-io 617  ax-5 1315  ax-7 1316  ax-gen 1317  ax-ie1 1362  ax-ie2 1363  ax-8 1377  ax-10 1378  ax-11 1379  ax-i12 1380  ax-bnd 1381  ax-4 1382  ax-17 1401  ax-i9 1405  ax-ial 1410  ax-i5r 1411  ax-ext 2004
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1329  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-rab 2291  df-v 2535  df-dif 2898  df-nul 3203
This theorem is referenced by: (None)
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