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Theorem uniin 3591
Description: The class union of the intersection of two classes. Exercise 4.12(n) of [Mendelson] p. 235. (Contributed by NM, 4-Dec-2003.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
Assertion
Ref Expression
uniin (AB) ⊆ ( A B)

Proof of Theorem uniin
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 19.40 1519 . . . 4 (y((x y y A) (x y y B)) → (y(x y y A) y(x y y B)))
2 elin 3120 . . . . . . 7 (y (AB) ↔ (y A y B))
32anbi2i 430 . . . . . 6 ((x y y (AB)) ↔ (x y (y A y B)))
4 anandi 524 . . . . . 6 ((x y (y A y B)) ↔ ((x y y A) (x y y B)))
53, 4bitri 173 . . . . 5 ((x y y (AB)) ↔ ((x y y A) (x y y B)))
65exbii 1493 . . . 4 (y(x y y (AB)) ↔ y((x y y A) (x y y B)))
7 eluni 3574 . . . . 5 (x Ay(x y y A))
8 eluni 3574 . . . . 5 (x By(x y y B))
97, 8anbi12i 433 . . . 4 ((x A x B) ↔ (y(x y y A) y(x y y B)))
101, 6, 93imtr4i 190 . . 3 (y(x y y (AB)) → (x A x B))
11 eluni 3574 . . 3 (x (AB) ↔ y(x y y (AB)))
12 elin 3120 . . 3 (x ( A B) ↔ (x A x B))
1310, 11, 123imtr4i 190 . 2 (x (AB) → x ( A B))
1413ssriv 2943 1 (AB) ⊆ ( A B)
Colors of variables: wff set class
Syntax hints:   wa 97  wex 1378   wcel 1390  cin 2910  wss 2911   cuni 3571
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-v 2553  df-in 2918  df-ss 2925  df-uni 3572
This theorem is referenced by: (None)
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