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Theorem iinss2 4792
 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 3417 . . . 4 𝑦 ∈ V
2 eliin 4745 . . . 4 (𝑦 ∈ V → (𝑦 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴 𝑦𝐵))
31, 2ax-mp 5 . . 3 (𝑦 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴 𝑦𝐵)
4 rsp 3138 . . . 4 (∀𝑥𝐴 𝑦𝐵 → (𝑥𝐴𝑦𝐵))
54com12 32 . . 3 (𝑥𝐴 → (∀𝑥𝐴 𝑦𝐵𝑦𝐵))
63, 5syl5bi 234 . 2 (𝑥𝐴 → (𝑦 𝑥𝐴 𝐵𝑦𝐵))
76ssrdv 3833 1 (𝑥𝐴 𝑥𝐴 𝐵𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 198   ∈ wcel 2166  ∀wral 3117  Vcvv 3414   ⊆ wss 3798  ∩ ciin 4741 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-ext 2803 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-v 3416  df-in 3805  df-ss 3812  df-iin 4743 This theorem is referenced by:  dmiin  5602  gruiin  9947  txtube  21814  iooiinicc  40564  iooiinioc  40578  meaiininclem  41494  smfsuplem1  41811  smfsuplem3  41813  smflimsuplem2  41821
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