ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  iinuniss GIF version

Theorem iinuniss 3779
Description: A relationship involving union and indexed intersection. Exercise 23 of [Enderton] p. 33 but with equality changed to subset. (Contributed by Jim Kingdon, 19-Aug-2018.)
Assertion
Ref Expression
iinuniss (𝐴 𝐵) ⊆ 𝑥𝐵 (𝐴𝑥)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem iinuniss
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 r19.32vr 2508 . . . 4 ((𝑦𝐴 ∨ ∀𝑥𝐵 𝑦𝑥) → ∀𝑥𝐵 (𝑦𝐴𝑦𝑥))
2 vex 2614 . . . . . 6 𝑦 ∈ V
32elint2 3664 . . . . 5 (𝑦 𝐵 ↔ ∀𝑥𝐵 𝑦𝑥)
43orbi2i 712 . . . 4 ((𝑦𝐴𝑦 𝐵) ↔ (𝑦𝐴 ∨ ∀𝑥𝐵 𝑦𝑥))
5 elun 3124 . . . . 5 (𝑦 ∈ (𝐴𝑥) ↔ (𝑦𝐴𝑦𝑥))
65ralbii 2378 . . . 4 (∀𝑥𝐵 𝑦 ∈ (𝐴𝑥) ↔ ∀𝑥𝐵 (𝑦𝐴𝑦𝑥))
71, 4, 63imtr4i 199 . . 3 ((𝑦𝐴𝑦 𝐵) → ∀𝑥𝐵 𝑦 ∈ (𝐴𝑥))
87ss2abi 3076 . 2 {𝑦 ∣ (𝑦𝐴𝑦 𝐵)} ⊆ {𝑦 ∣ ∀𝑥𝐵 𝑦 ∈ (𝐴𝑥)}
9 df-un 2987 . 2 (𝐴 𝐵) = {𝑦 ∣ (𝑦𝐴𝑦 𝐵)}
10 df-iin 3702 . 2 𝑥𝐵 (𝐴𝑥) = {𝑦 ∣ ∀𝑥𝐵 𝑦 ∈ (𝐴𝑥)}
118, 9, 103sstr4i 3048 1 (𝐴 𝐵) ⊆ 𝑥𝐵 (𝐴𝑥)
Colors of variables: wff set class
Syntax hints:  wo 662  wcel 1434  {cab 2069  wral 2353  cun 2981  wss 2983   cint 3657   ciin 3700
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-v 2613  df-un 2987  df-in 2989  df-ss 2996  df-int 3658  df-iin 3702
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator