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Theorem rab0 3571
 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 equid 1676 . . . . 5 x = x
2 noel 3554 . . . . . 6 ¬ x
32intnanr 881 . . . . 5 ¬ (x φ)
41, 32th 230 . . . 4 (x = x ↔ ¬ (x φ))
54con2bii 322 . . 3 ((x φ) ↔ ¬ x = x)
65abbii 2465 . 2 {x (x φ)} = {x ¬ x = x}
7 df-rab 2623 . 2 {x φ} = {x (x φ)}
8 dfnul2 3552 . 2 = {x ¬ x = x}
96, 7, 83eqtr4i 2383 1 {x φ} =
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ∧ wa 358   = wceq 1642   ∈ wcel 1710  {cab 2339  {crab 2618  ∅c0 3550 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-rab 2623  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-dif 3215  df-nul 3551 This theorem is referenced by: (None)
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