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Theorem iinss 4144
 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 2961 . . . 4
2 eliin 4100 . . . 4
31, 2ax-mp 5 . . 3
4 ssel 3344 . . . . 5
54reximi 2815 . . . 4
6 r19.36av 2858 . . . 4
75, 6syl 16 . . 3
83, 7syl5bi 210 . 2
98ssrdv 3356 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 178   wcel 1726  wral 2707  wrex 2708  cvv 2958   wss 3322  ciin 4096 This theorem is referenced by:  riinn0  4168  reliin  4999  cnviin  5412  iiner  6979  scott0  7815  cfslb  8151  ptbasfi  17618  iscmet3  19251  fnemeet1  26409  pmapglb2N  30642  pmapglb2xN  30643 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2712  df-rex 2713  df-v 2960  df-in 3329  df-ss 3336  df-iin 4098
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