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

Theorem iinuniss 3841
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 2537 . . . 4 ((𝑦𝐴 ∨ ∀𝑥𝐵 𝑦𝑥) → ∀𝑥𝐵 (𝑦𝐴𝑦𝑥))
2 vex 2644 . . . . . 6 𝑦 ∈ V
32elint2 3725 . . . . 5 (𝑦 𝐵 ↔ ∀𝑥𝐵 𝑦𝑥)
43orbi2i 720 . . . 4 ((𝑦𝐴𝑦 𝐵) ↔ (𝑦𝐴 ∨ ∀𝑥𝐵 𝑦𝑥))
5 elun 3164 . . . . 5 (𝑦 ∈ (𝐴𝑥) ↔ (𝑦𝐴𝑦𝑥))
65ralbii 2400 . . . 4 (∀𝑥𝐵 𝑦 ∈ (𝐴𝑥) ↔ ∀𝑥𝐵 (𝑦𝐴𝑦𝑥))
71, 4, 63imtr4i 200 . . 3 ((𝑦𝐴𝑦 𝐵) → ∀𝑥𝐵 𝑦 ∈ (𝐴𝑥))
87ss2abi 3116 . 2 {𝑦 ∣ (𝑦𝐴𝑦 𝐵)} ⊆ {𝑦 ∣ ∀𝑥𝐵 𝑦 ∈ (𝐴𝑥)}
9 df-un 3025 . 2 (𝐴 𝐵) = {𝑦 ∣ (𝑦𝐴𝑦 𝐵)}
10 df-iin 3763 . 2 𝑥𝐵 (𝐴𝑥) = {𝑦 ∣ ∀𝑥𝐵 𝑦 ∈ (𝐴𝑥)}
118, 9, 103sstr4i 3088 1 (𝐴 𝐵) ⊆ 𝑥𝐵 (𝐴𝑥)
Colors of variables: wff set class
Syntax hints:  wo 670  wcel 1448  {cab 2086  wral 2375  cun 3019  wss 3021   cint 3718   ciin 3761
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082
This theorem depends on definitions:  df-bi 116  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ral 2380  df-v 2643  df-un 3025  df-in 3027  df-ss 3034  df-int 3719  df-iin 3763
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator