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Theorem iunxsng 4044
 Description: A singleton index picks out an instance of an indexed union's argument. (Contributed by Mario Carneiro, 25-Jun-2016.)
Hypothesis
Ref Expression
iunxsng.1 (x = AB = C)
Assertion
Ref Expression
iunxsng (A Vx {A}B = C)
Distinct variable groups:   x,A   x,C
Allowed substitution hints:   B(x)   V(x)

Proof of Theorem iunxsng
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 eliun 3973 . . 3 (y x {A}Bx {A}y B)
2 iunxsng.1 . . . . 5 (x = AB = C)
32eleq2d 2420 . . . 4 (x = A → (y By C))
43rexsng 3766 . . 3 (A V → (x {A}y By C))
51, 4syl5bb 248 . 2 (A V → (y x {A}By C))
65eqrdv 2351 1 (A Vx {A}B = C)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1642   ∈ wcel 1710  ∃wrex 2615  {csn 3737  ∪ciun 3969 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-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:  iunxsn  4045
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