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Theorem iinssiun 4891
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 4378 . . . 4 ((𝐴 ≠ ∅ ∧ ∀𝑥𝐴 𝑦𝐵) → ∃𝑥𝐴 𝑦𝐵)
21ex 416 . . 3 (𝐴 ≠ ∅ → (∀𝑥𝐴 𝑦𝐵 → ∃𝑥𝐴 𝑦𝐵))
3 eliin 4883 . . . 4 (𝑦 ∈ V → (𝑦 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴 𝑦𝐵))
43elv 3403 . . 3 (𝑦 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴 𝑦𝐵)
5 eliun 4882 . . 3 (𝑦 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴 𝑦𝐵)
62, 4, 53imtr4g 299 . 2 (𝐴 ≠ ∅ → (𝑦 𝑥𝐴 𝐵𝑦 𝑥𝐴 𝐵))
76ssrdv 3881 1 (𝐴 ≠ ∅ → 𝑥𝐴 𝐵 𝑥𝐴 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wcel 2113  wne 2934  wral 3053  wrex 3054  Vcvv 3397  wss 3841  c0 4209   ciun 4878   ciin 4879
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-ext 2710
This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1545  df-fal 1555  df-ex 1787  df-sb 2074  df-clab 2717  df-cleq 2730  df-clel 2811  df-ne 2935  df-ral 3058  df-rex 3059  df-v 3399  df-dif 3844  df-in 3848  df-ss 3858  df-nul 4210  df-iun 4880  df-iin 4881
This theorem is referenced by:  subdrgint  19694
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