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Theorem iinss2 4604
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 3234 . . . 4 𝑦 ∈ V
2 eliin 4557 . . . 4 (𝑦 ∈ V → (𝑦 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴 𝑦𝐵))
31, 2ax-mp 5 . . 3 (𝑦 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴 𝑦𝐵)
4 rsp 2958 . . . 4 (∀𝑥𝐴 𝑦𝐵 → (𝑥𝐴𝑦𝐵))
54com12 32 . . 3 (𝑥𝐴 → (∀𝑥𝐴 𝑦𝐵𝑦𝐵))
63, 5syl5bi 232 . 2 (𝑥𝐴 → (𝑦 𝑥𝐴 𝐵𝑦𝐵))
76ssrdv 3642 1 (𝑥𝐴 𝑥𝐴 𝐵𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wcel 2030  wral 2941  Vcvv 3231  wss 3607   ciin 4553
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-v 3233  df-in 3614  df-ss 3621  df-iin 4555
This theorem is referenced by:  dmiin  5401  gruiin  9670  txtube  21491  iooiinicc  40087  iooiinioc  40101  meaiininclem  41021  smfsuplem1  41338  smfsuplem3  41340  smflimsuplem2  41348
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