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Theorem dfnul3 3553
 Description: Alternate definition of the empty set. (Contributed by NM, 25-Mar-2004.)
Assertion
Ref Expression
dfnul3 = {x A ¬ x A}

Proof of Theorem dfnul3
StepHypRef Expression
1 pm3.24 852 . . . . 5 ¬ (x A ¬ x A)
2 eqid 2353 . . . . 5 x = x
31, 22th 230 . . . 4 (¬ (x A ¬ x A) ↔ x = x)
43con1bii 321 . . 3 x = x ↔ (x A ¬ x A))
54abbii 2465 . 2 {x ¬ x = x} = {x (x A ¬ x A)}
6 dfnul2 3552 . 2 = {x ¬ x = x}
7 df-rab 2623 . 2 {x A ¬ x A} = {x (x A ¬ x A)}
85, 6, 73eqtr4i 2383 1 = {x A ¬ x A}
 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-mp 5  ax-1 6  ax-2 7  ax-3 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:  difidALT  3619
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