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Theorem iinss2 4987
Description: An indexed intersection is included in any of its members. (Contributed by FL, 15-Oct-2012.)
Assertion
Ref Expression
iinss2 (𝑥𝐴 𝑥𝐴 𝐵𝐵)

Proof of Theorem iinss2
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 eliin 4929 . . . 4 (𝑦 ∈ V → (𝑦 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴 𝑦𝐵))
21elv 3438 . . 3 (𝑦 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴 𝑦𝐵)
3 rsp 3131 . . . 4 (∀𝑥𝐴 𝑦𝐵 → (𝑥𝐴𝑦𝐵))
43com12 32 . . 3 (𝑥𝐴 → (∀𝑥𝐴 𝑦𝐵𝑦𝐵))
52, 4syl5bi 241 . 2 (𝑥𝐴 → (𝑦 𝑥𝐴 𝐵𝑦𝐵))
65ssrdv 3927 1 (𝑥𝐴 𝑥𝐴 𝐵𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wcel 2106  wral 3064  Vcvv 3432  wss 3887   ciin 4925
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-12 2171  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-v 3434  df-in 3894  df-ss 3904  df-iin 4927
This theorem is referenced by:  dmiin  5862  gruiin  10566  txtube  22791  iooiinicc  43080  iooiinioc  43094  meaiininclem  44024  smfsuplem1  44344  smfsuplem3  44346  smflimsuplem2  44354
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