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Theorem rab0 3243
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 3225 . . . . 5 ¬ x
21intnanr 839 . . . 4 ¬ (x φ)
3 equid 1589 . . . . 5 x = x
43notnoti 574 . . . 4 ¬ ¬ x = x
52, 42false 617 . . 3 ((x φ) ↔ ¬ x = x)
65abbii 2153 . 2 {x ∣ (x φ)} = {x ∣ ¬ x = x}
7 df-rab 2312 . 2 {x ∅ ∣ φ} = {x ∣ (x φ)}
8 dfnul2 3223 . 2 ∅ = {x ∣ ¬ x = x}
96, 7, 83eqtr4i 2070 1 {x ∅ ∣ φ} = ∅
Colors of variables: wff set class
Syntax hints:  ¬ wn 3   wa 97   = wceq 1243   wcel 1393  {cab 2026  {crab 2307  c0 3221
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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rab 2312  df-v 2556  df-dif 2917  df-nul 3222
This theorem is referenced by: (None)
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