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Theorem iunxsnf 39754
Description: A singleton index picks out an instance of an indexed union's argument. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
Hypotheses
Ref Expression
iunxsnf.1 𝑥𝐶
iunxsnf.2 𝐴 ∈ V
iunxsnf.3 (𝑥 = 𝐴𝐵 = 𝐶)
Assertion
Ref Expression
iunxsnf 𝑥 ∈ {𝐴}𝐵 = 𝐶
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝐵(𝑥)   𝐶(𝑥)

Proof of Theorem iunxsnf
StepHypRef Expression
1 iunxsnf.2 . 2 𝐴 ∈ V
2 iunxsnf.1 . . 3 𝑥𝐶
3 iunxsnf.3 . . 3 (𝑥 = 𝐴𝐵 = 𝐶)
42, 3iunxsngf2 39751 . 2 (𝐴 ∈ V → 𝑥 ∈ {𝐴}𝐵 = 𝐶)
51, 4ax-mp 5 1 𝑥 ∈ {𝐴}𝐵 = 𝐶
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1631  wcel 2145  wnfc 2900  Vcvv 3351  {csn 4316   ciun 4654
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-v 3353  df-sbc 3588  df-sn 4317  df-iun 4656
This theorem is referenced by:  fiiuncl  39755  iunp1  39756  sge0iunmptlemfi  41147
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