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Theorem iinss 4381
 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:   ,

Proof of Theorem iinss
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 vex 3112 . . . 4
2 eliin 4336 . . . 4
31, 2ax-mp 5 . . 3
4 ssel 3497 . . . . 5
54reximi 2925 . . . 4
6 r19.36v 3005 . . . 4
75, 6syl 16 . . 3
83, 7syl5bi 217 . 2
98ssrdv 3509 1
 Colors of variables: wff setvar class Syntax hints:  ->wi 4  <->wb 184  e.wcel 1818  A.wral 2807  E.wrex 2808   cvv 3109  C_wss 3475  |^|_ciin 4331 This theorem is referenced by:  riinn0  4405  reliin  5129  cnviin  5549  iiner  7402  scott0  8325  cfslb  8667  ptbasfi  20082  iscmet3  21732  fnemeet1  30184  pmapglb2N  35495  pmapglb2xN  35496 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435 This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ral 2812  df-rex 2813  df-v 3111  df-in 3482  df-ss 3489  df-iin 4333
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