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Theorem ssiinf 3862
 Description: Subset theorem for an indexed intersection. (Contributed by FL, 15-Oct-2012.) (Proof shortened by Mario Carneiro, 14-Oct-2016.)
Hypothesis
Ref Expression
ssiinf.1
Assertion
Ref Expression
ssiinf

Proof of Theorem ssiinf
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 vex 2689 . . . . 5
2 eliin 3818 . . . . 5
31, 2ax-mp 5 . . . 4
43ralbii 2441 . . 3
5 ssiinf.1 . . . 4
6 nfcv 2281 . . . 4
75, 6ralcomf 2592 . . 3
84, 7bitri 183 . 2
9 dfss3 3087 . 2
10 dfss3 3087 . . 3
1110ralbii 2441 . 2
128, 9, 113bitr4i 211 1
 Colors of variables: wff set class Syntax hints:   wb 104   wcel 1480  wnfc 2268  wral 2416  cvv 2686   wss 3071  ciin 3814 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 2121 This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-v 2688  df-in 3077  df-ss 3084  df-iin 3816 This theorem is referenced by:  ssiin  3863  dmiin  4785
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