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Theorem iinss 4293
 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 3025 . . . 4
2 eliin 4248 . . . 4
31, 2ax-mp 5 . . 3
4 ssel 3401 . . . . 5
54reximi 2832 . . . 4
6 r19.36v 2915 . . . 4
75, 6syl 17 . . 3
83, 7syl5bi 220 . 2
98ssrdv 3413 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 187   wcel 1872  wral 2714  wrex 2715  cvv 3022   wss 3379  ciin 4243 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408 This theorem depends on definitions:  df-bi 188  df-an 372  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ral 2719  df-rex 2720  df-v 3024  df-in 3386  df-ss 3393  df-iin 4245 This theorem is referenced by:  riinn0  4317  reliin  4917  cnviin  5335  iiner  7390  scott0  8309  cfslb  8647  ptbasfi  20538  iscmet3  22205  fnemeet1  30971  pmapglb2N  33248  pmapglb2xN  33249
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