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Theorem iunxsn 3855
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 3854 . 2 (𝐴 ∈ V → 𝑥 ∈ {𝐴}𝐵 = 𝐶)
41, 3ax-mp 7 1 𝑥 ∈ {𝐴}𝐵 = 𝐶
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1314  wcel 1463  Vcvv 2657  {csn 3493   ciun 3779
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ral 2395  df-rex 2396  df-v 2659  df-sbc 2879  df-sn 3499  df-iun 3781
This theorem is referenced by:  iunsuc  4302  fsum2dlemstep  11092  fsumiun  11135
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