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Theorem iinss 4017
 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 2862 . . . 4
2 eliin 3974 . . . 4
31, 2ax-mp 8 . . 3
4 ssel 3267 . . . . 5
54reximi 2721 . . . 4
6 r19.36av 2759 . . . 4
75, 6syl 15 . . 3
83, 7syl5bi 208 . 2
98ssrdv 3278 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 176   wcel 1710  wral 2614  wrex 2615  cvv 2859   wss 3257  ciin 3970 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ral 2619  df-rex 2620  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-ss 3259  df-iin 3972 This theorem is referenced by:  riinn0  4040
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