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Theorem iinssiun 4937
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 4425 . . . 4 ((𝐴 ≠ ∅ ∧ ∀𝑥𝐴 𝑦𝐵) → ∃𝑥𝐴 𝑦𝐵)
21ex 413 . . 3 (𝐴 ≠ ∅ → (∀𝑥𝐴 𝑦𝐵 → ∃𝑥𝐴 𝑦𝐵))
3 eliin 4929 . . . 4 (𝑦 ∈ V → (𝑦 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴 𝑦𝐵))
43elv 3438 . . 3 (𝑦 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴 𝑦𝐵)
5 eliun 4928 . . 3 (𝑦 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴 𝑦𝐵)
62, 4, 53imtr4g 296 . 2 (𝐴 ≠ ∅ → (𝑦 𝑥𝐴 𝐵𝑦 𝑥𝐴 𝐵))
76ssrdv 3927 1 (𝐴 ≠ ∅ → 𝑥𝐴 𝐵 𝑥𝐴 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wcel 2106  wne 2943  wral 3064  wrex 3065  Vcvv 3432  wss 3887  c0 4256   ciun 4924   ciin 4925
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ne 2944  df-ral 3069  df-rex 3070  df-v 3434  df-dif 3890  df-in 3894  df-ss 3904  df-nul 4257  df-iun 4926  df-iin 4927
This theorem is referenced by:  subdrgint  20071
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