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Theorem elimaksn 4283
 Description: Membership in a Kuratowski image of a singleton. (Contributed by SF, 4-Feb-2015.)
Hypotheses
Ref Expression
elimaksn.1 B V
elimaksn.2 C V
Assertion
Ref Expression
elimaksn (C (Ak {B}) ↔ ⟪B, C A)

Proof of Theorem elimaksn
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 elimaksn.2 . . 3 C V
21elimak 4259 . 2 (C (Ak {B}) ↔ x {B}⟪x, C A)
3 elimaksn.1 . . 3 B V
4 opkeq1 4059 . . . 4 (x = B → ⟪x, C⟫ = ⟪B, C⟫)
54eleq1d 2419 . . 3 (x = B → (⟪x, C A ↔ ⟪B, C A))
63, 5rexsn 3768 . 2 (x {B}⟪x, C A ↔ ⟪B, C A)
72, 6bitri 240 1 (C (Ak {B}) ↔ ⟪B, C A)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176   = wceq 1642   ∈ wcel 1710  ∃wrex 2615  Vcvv 2859  {csn 3737  ⟪copk 4057   “k cimak 4179 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-3an 936  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-rex 2620  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-un 3214  df-sn 3741  df-pr 3742  df-opk 4058  df-imak 4189 This theorem is referenced by:  preaddccan2lem1  4454
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