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

Theorem iinss 3859
 Description: Subset implication for an indexed intersection. (Contributed by NM, 15-Oct-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
Assertion
Ref Expression
iinss
Distinct variable group:   ,
Allowed substitution hints:   ()   ()

Proof of Theorem iinss
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 vex 2684 . . . 4
2 eliin 3813 . . . 4
31, 2ax-mp 5 . . 3
4 ssel 3086 . . . . 5
54reximi 2527 . . . 4
6 r19.36av 2580 . . . 4
75, 6syl 14 . . 3
83, 7syl5bi 151 . 2
98ssrdv 3098 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 104   wcel 1480  wral 2414  wrex 2415  cvv 2681   wss 3066  ciin 3809 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119 This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-in 3072  df-ss 3079  df-iin 3811 This theorem is referenced by:  riinm  3880  reliin  4656  cnviinm  5075  iinerm  6494
 Copyright terms: Public domain W3C validator