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Theorem iunxsn 3925
 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 3924 . 2 (𝐴 ∈ V → 𝑥 ∈ {𝐴}𝐵 = 𝐶)
41, 3ax-mp 5 1 𝑥 ∈ {𝐴}𝐵 = 𝐶
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1335   ∈ wcel 2128  Vcvv 2712  {csn 3560  ∪ ciun 3849 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-sbc 2938  df-sn 3566  df-iun 3851 This theorem is referenced by:  iunsuc  4380  fsum2dlemstep  11324  fsumiun  11367  fprod2dlemstep  11512
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