Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  iunxsn GIF version

Theorem iunxsn 3724
 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 3723 . 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 1242   ∈ wcel 1390  Vcvv 2551  {csn 3367  ∪ ciun 3648 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-sn 3373  df-iun 3650 This theorem is referenced by:  iunsuc  4123
 Copyright terms: Public domain W3C validator