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Theorem iinssiun 5004
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 4490 . . . 4 ((𝐴 ≠ ∅ ∧ ∀𝑥𝐴 𝑦𝐵) → ∃𝑥𝐴 𝑦𝐵)
21ex 412 . . 3 (𝐴 ≠ ∅ → (∀𝑥𝐴 𝑦𝐵 → ∃𝑥𝐴 𝑦𝐵))
3 eliin 4996 . . . 4 (𝑦 ∈ V → (𝑦 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴 𝑦𝐵))
43elv 3475 . . 3 (𝑦 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴 𝑦𝐵)
5 eliun 4995 . . 3 (𝑦 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴 𝑦𝐵)
62, 4, 53imtr4g 296 . 2 (𝐴 ≠ ∅ → (𝑦 𝑥𝐴 𝐵𝑦 𝑥𝐴 𝐵))
76ssrdv 3984 1 (𝐴 ≠ ∅ → 𝑥𝐴 𝐵 𝑥𝐴 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wcel 2099  wne 2935  wral 3056  wrex 3065  Vcvv 3469  wss 3944  c0 4318   ciun 4991   ciin 4992
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2698
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2705  df-cleq 2719  df-clel 2805  df-ne 2936  df-ral 3057  df-rex 3066  df-v 3471  df-dif 3947  df-in 3951  df-ss 3961  df-nul 4319  df-iun 4993  df-iin 4994
This theorem is referenced by:  subdrgint  20680
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