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Theorem iinss2 3736
 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 2577 . . . . 5 𝑦 ∈ V
2 eliin 3689 . . . . 5 (𝑦 ∈ V → (𝑦 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴 𝑦𝐵))
31, 2ax-mp 7 . . . 4 (𝑦 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴 𝑦𝐵)
4 rsp 2386 . . . 4 (∀𝑥𝐴 𝑦𝐵 → (𝑥𝐴𝑦𝐵))
53, 4sylbi 118 . . 3 (𝑦 𝑥𝐴 𝐵 → (𝑥𝐴𝑦𝐵))
65com12 30 . 2 (𝑥𝐴 → (𝑦 𝑥𝐴 𝐵𝑦𝐵))
76ssrdv 2978 1 (𝑥𝐴 𝑥𝐴 𝐵𝐵)
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 102   ∈ wcel 1409  ∀wral 2323  Vcvv 2574   ⊆ wss 2944  ∩ ciin 3685 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038 This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-v 2576  df-in 2951  df-ss 2958  df-iin 3687 This theorem is referenced by:  dmiin  4607
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