Mathbox for Glauco Siliprandi < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  iunxsnf Structured version   Visualization version   GIF version

Theorem iunxsnf 40045
 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 40042 . 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 1656   ∈ wcel 2164  Ⅎwnfc 2956  Vcvv 3414  {csn 4399  ∪ ciun 4742 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-v 3416  df-sbc 3663  df-sn 4400  df-iun 4744 This theorem is referenced by:  fiiuncl  40046  iunp1  40047  sge0iunmptlemfi  41415
 Copyright terms: Public domain W3C validator