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Theorem uniin 3911
 Description: The class union of the intersection of two classes. Exercise 4.12(n) of [Mendelson] p. 235. See uniinqs in set.mm for a condition where equality holds. (Contributed by NM, 4-Dec-2003.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
Assertion
Ref Expression
uniin (AB) (AB)

Proof of Theorem uniin
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 19.40 1609 . . . 4 (y((x y y A) (x y y B)) → (y(x y y A) y(x y y B)))
2 elin 3219 . . . . . . 7 (y (AB) ↔ (y A y B))
32anbi2i 675 . . . . . 6 ((x y y (AB)) ↔ (x y (y A y B)))
4 anandi 801 . . . . . 6 ((x y (y A y B)) ↔ ((x y y A) (x y y B)))
53, 4bitri 240 . . . . 5 ((x y y (AB)) ↔ ((x y y A) (x y y B)))
65exbii 1582 . . . 4 (y(x y y (AB)) ↔ y((x y y A) (x y y B)))
7 eluni 3894 . . . . 5 (x Ay(x y y A))
8 eluni 3894 . . . . 5 (x By(x y y B))
97, 8anbi12i 678 . . . 4 ((x A x B) ↔ (y(x y y A) y(x y y B)))
101, 6, 93imtr4i 257 . . 3 (y(x y y (AB)) → (x A x B))
11 eluni 3894 . . 3 (x (AB) ↔ y(x y y (AB)))
12 elin 3219 . . 3 (x (AB) ↔ (x A x B))
1310, 11, 123imtr4i 257 . 2 (x (AB) → x (AB))
1413ssriv 3277 1 (AB) (AB)
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 358  ∃wex 1541   ∈ wcel 1710   ∩ cin 3208   ⊆ wss 3257  ∪cuni 3891 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-ss 3259  df-uni 3892 This theorem is referenced by: (None)
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