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Theorem iunxsn 4045
 Description: A singleton index picks out an instance of an indexed union's argument. (Contributed by NM, 26-Mar-2004.) (Proof shortened by Mario Carneiro, 25-Jun-2016.)
Hypotheses
Ref Expression
iunxsn.1 A V
iunxsn.2 (x = AB = C)
Assertion
Ref Expression
iunxsn x {A}B = C
Distinct variable groups:   x,A   x,C
Allowed substitution hint:   B(x)

Proof of Theorem iunxsn
StepHypRef Expression
1 iunxsn.1 . 2 A V
2 iunxsn.2 . . 3 (x = AB = C)
32iunxsng 4044 . 2 (A V → x {A}B = C)
41, 3ax-mp 8 1 x {A}B = C
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1642   ∈ wcel 1710  Vcvv 2859  {csn 3737  ∪ciun 3969 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-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ral 2619  df-rex 2620  df-v 2861  df-sbc 3047  df-sn 3741  df-iun 3971 This theorem is referenced by: (None)
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