Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  iinssiun Structured version   Visualization version   GIF version

Theorem iinssiun 4923
 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 4438 . . . 4 ((𝐴 ≠ ∅ ∧ ∀𝑥𝐴 𝑦𝐵) → ∃𝑥𝐴 𝑦𝐵)
21ex 415 . . 3 (𝐴 ≠ ∅ → (∀𝑥𝐴 𝑦𝐵 → ∃𝑥𝐴 𝑦𝐵))
3 eliin 4915 . . . 4 (𝑦 ∈ V → (𝑦 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴 𝑦𝐵))
43elv 3498 . . 3 (𝑦 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴 𝑦𝐵)
5 eliun 4914 . . 3 (𝑦 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴 𝑦𝐵)
62, 4, 53imtr4g 298 . 2 (𝐴 ≠ ∅ → (𝑦 𝑥𝐴 𝐵𝑦 𝑥𝐴 𝐵))
76ssrdv 3971 1 (𝐴 ≠ ∅ → 𝑥𝐴 𝐵 𝑥𝐴 𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 208   ∈ wcel 2107   ≠ wne 3014  ∀wral 3136  ∃wrex 3137  Vcvv 3493   ⊆ wss 3934  ∅c0 4289  ∪ ciun 4910  ∩ ciin 4911 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-v 3495  df-dif 3937  df-in 3941  df-ss 3950  df-nul 4290  df-iun 4912  df-iin 4913 This theorem is referenced by:  subdrgint  19574
 Copyright terms: Public domain W3C validator