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Theorem iinssiun 5009
Description: An indexed intersection is a subset of the corresponding indexed union. (Contributed by Thierry Arnoux, 31-Dec-2021.)
Assertion
Ref Expression
iinssiun (𝐴 ≠ ∅ → 𝑥𝐴 𝐵 𝑥𝐴 𝐵)
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem iinssiun
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 r19.2z 4500 . . . 4 ((𝐴 ≠ ∅ ∧ ∀𝑥𝐴 𝑦𝐵) → ∃𝑥𝐴 𝑦𝐵)
21ex 412 . . 3 (𝐴 ≠ ∅ → (∀𝑥𝐴 𝑦𝐵 → ∃𝑥𝐴 𝑦𝐵))
3 eliin 5000 . . . 4 (𝑦 ∈ V → (𝑦 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴 𝑦𝐵))
43elv 3482 . . 3 (𝑦 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴 𝑦𝐵)
5 eliun 4999 . . 3 (𝑦 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴 𝑦𝐵)
62, 4, 53imtr4g 296 . 2 (𝐴 ≠ ∅ → (𝑦 𝑥𝐴 𝐵𝑦 𝑥𝐴 𝐵))
76ssrdv 4000 1 (𝐴 ≠ ∅ → 𝑥𝐴 𝐵 𝑥𝐴 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wcel 2105  wne 2937  wral 3058  wrex 3067  Vcvv 3477  wss 3962  c0 4338   ciun 4995   ciin 4996
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1539  df-fal 1549  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-ne 2938  df-ral 3059  df-rex 3068  df-v 3479  df-dif 3965  df-ss 3979  df-nul 4339  df-iun 4997  df-iin 4998
This theorem is referenced by:  subdrgint  20820
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