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Theorem iinuniss 3727
 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) ⊆ x B (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 2452 . . . 4 ((y A x B y x) → x B (y A y x))
2 vex 2554 . . . . . 6 y V
32elint2 3612 . . . . 5 (y Bx B y x)
43orbi2i 678 . . . 4 ((y A y B) ↔ (y A x B y x))
5 elun 3078 . . . . 5 (y (Ax) ↔ (y A y x))
65ralbii 2324 . . . 4 (x B y (Ax) ↔ x B (y A y x))
71, 4, 63imtr4i 190 . . 3 ((y A y B) → x B y (Ax))
87ss2abi 3006 . 2 {y ∣ (y A y B)} ⊆ {yx B y (Ax)}
9 df-un 2916 . 2 (A B) = {y ∣ (y A y B)}
10 df-iin 3650 . 2 x B (Ax) = {yx B y (Ax)}
118, 9, 103sstr4i 2978 1 (A B) ⊆ x B (Ax)
 Colors of variables: wff set class Syntax hints:   ∨ wo 628   ∈ wcel 1390  {cab 2023  ∀wral 2300   ∪ cun 2909   ⊆ wss 2911  ∩ cint 3605  ∩ ciin 3648 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-int 3606  df-iin 3650 This theorem is referenced by: (None)
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