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Theorem iunxsnf 45069
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, 3iunxsngf 5092 . 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 1540  wcel 2108  wnfc 2890  Vcvv 3480  {csn 4626   ciun 4991
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-12 2177  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1089  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-rex 3071  df-v 3482  df-sbc 3789  df-sn 4627  df-iun 4993
This theorem is referenced by:  fiiuncl  45070  iunp1  45071  sge0iunmptlemfi  46428
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