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Theorem iinss2 3901
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 vex 2715 . . . . 5 𝑦 ∈ V
2 eliin 3854 . . . . 5 (𝑦 ∈ V → (𝑦 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴 𝑦𝐵))
31, 2ax-mp 5 . . . 4 (𝑦 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴 𝑦𝐵)
4 rsp 2504 . . . 4 (∀𝑥𝐴 𝑦𝐵 → (𝑥𝐴𝑦𝐵))
53, 4sylbi 120 . . 3 (𝑦 𝑥𝐴 𝐵 → (𝑥𝐴𝑦𝐵))
65com12 30 . 2 (𝑥𝐴 → (𝑦 𝑥𝐴 𝐵𝑦𝐵))
76ssrdv 3134 1 (𝑥𝐴 𝑥𝐴 𝐵𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104  wcel 2128  wral 2435  Vcvv 2712  wss 3102   ciin 3850
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 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-v 2714  df-in 3108  df-ss 3115  df-iin 3852
This theorem is referenced by:  dmiin  4831
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