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Theorem iinss2 4972
 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 4915 . . . 4 (𝑦 ∈ V → (𝑦 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴 𝑦𝐵))
21elv 3498 . . 3 (𝑦 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴 𝑦𝐵)
3 rsp 3203 . . . 4 (∀𝑥𝐴 𝑦𝐵 → (𝑥𝐴𝑦𝐵))
43com12 32 . . 3 (𝑥𝐴 → (∀𝑥𝐴 𝑦𝐵𝑦𝐵))
52, 4syl5bi 244 . 2 (𝑥𝐴 → (𝑦 𝑥𝐴 𝐵𝑦𝐵))
65ssrdv 3971 1 (𝑥𝐴 𝑥𝐴 𝐵𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 208   ∈ wcel 2108  ∀wral 3136  Vcvv 3493   ⊆ wss 3934  ∩ ciin 4911 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ral 3141  df-v 3495  df-in 3941  df-ss 3950  df-iin 4913 This theorem is referenced by:  dmiin  5818  gruiin  10224  txtube  22240  iooiinicc  41808  iooiinioc  41822  meaiininclem  42759  smfsuplem1  43076  smfsuplem3  43078  smflimsuplem2  43086
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