Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  iinxsng Unicode version

Theorem iinxsng 3978
 Description: A singleton index picks out an instance of an indexed intersection's argument. (Contributed by NM, 15-Jan-2012.) (Proof shortened by Mario Carneiro, 17-Nov-2016.)
Hypothesis
Ref Expression
iinxsng.1
Assertion
Ref Expression
iinxsng
Distinct variable groups:   ,   ,
Allowed substitution hints:   ()   ()

Proof of Theorem iinxsng
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 df-iin 3908 . 2
2 iinxsng.1 . . . . 5
32eleq2d 2350 . . . 4
43ralsng 3672 . . 3
54abbi1dv 2399 . 2
61, 5syl5eq 2327 1
 Colors of variables: wff set class Syntax hints:   wi 4   wceq 1623   wcel 1684  cab 2269  wral 2543  csn 3640  ciin 3906 This theorem is referenced by:  splintx  25049  polatN  30120 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-v 2790  df-sbc 2992  df-sn 3646  df-iin 3908
 Copyright terms: Public domain W3C validator