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Theorem elcomplg 3218
 Description: Membership in class complement. (Contributed by SF, 10-Jan-2015.)
Assertion
Ref Expression
elcomplg (A V → (A B ↔ ¬ A B))

Proof of Theorem elcomplg
StepHypRef Expression
1 df-compl 3212 . . 3 B = (BB)
21eleq2i 2417 . 2 (A BA (BB))
3 elning 3217 . . 3 (A V → (A (BB) ↔ (A B A B)))
4 df-nan 1288 . . . 4 ((A B A B) ↔ ¬ (A B A B))
5 anidm 625 . . . 4 ((A B A B) ↔ A B)
64, 5xchbinx 301 . . 3 ((A B A B) ↔ ¬ A B)
73, 6syl6bb 252 . 2 (A V → (A (BB) ↔ ¬ A B))
82, 7syl5bb 248 1 (A V → (A B ↔ ¬ A B))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∧ wa 358   ⊼ wnan 1287   ∈ wcel 1710   ⩃ cnin 3204   ∼ ccompl 3205 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-v 2861  df-nin 3211  df-compl 3212 This theorem is referenced by:  elin  3219  elun  3220  eldif  3221  elcompl  3225  nnadjoinpw  4521  nmembers1lem3  6270
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