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Theorem uniin 4931
Description: The class union of the intersection of two classes. Exercise 4.12(n) of [Mendelson] p. 235. See uniinqs 8818 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) ⊆ ( 𝐴 𝐵)

Proof of Theorem uniin
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 19.40 1882 . . . 4 (∃𝑦((𝑥𝑦𝑦𝐴) ∧ (𝑥𝑦𝑦𝐵)) → (∃𝑦(𝑥𝑦𝑦𝐴) ∧ ∃𝑦(𝑥𝑦𝑦𝐵)))
2 elin 3962 . . . . . . 7 (𝑦 ∈ (𝐴𝐵) ↔ (𝑦𝐴𝑦𝐵))
32anbi2i 621 . . . . . 6 ((𝑥𝑦𝑦 ∈ (𝐴𝐵)) ↔ (𝑥𝑦 ∧ (𝑦𝐴𝑦𝐵)))
4 anandi 674 . . . . . 6 ((𝑥𝑦 ∧ (𝑦𝐴𝑦𝐵)) ↔ ((𝑥𝑦𝑦𝐴) ∧ (𝑥𝑦𝑦𝐵)))
53, 4bitri 274 . . . . 5 ((𝑥𝑦𝑦 ∈ (𝐴𝐵)) ↔ ((𝑥𝑦𝑦𝐴) ∧ (𝑥𝑦𝑦𝐵)))
65exbii 1843 . . . 4 (∃𝑦(𝑥𝑦𝑦 ∈ (𝐴𝐵)) ↔ ∃𝑦((𝑥𝑦𝑦𝐴) ∧ (𝑥𝑦𝑦𝐵)))
7 eluni 4908 . . . . 5 (𝑥 𝐴 ↔ ∃𝑦(𝑥𝑦𝑦𝐴))
8 eluni 4908 . . . . 5 (𝑥 𝐵 ↔ ∃𝑦(𝑥𝑦𝑦𝐵))
97, 8anbi12i 626 . . . 4 ((𝑥 𝐴𝑥 𝐵) ↔ (∃𝑦(𝑥𝑦𝑦𝐴) ∧ ∃𝑦(𝑥𝑦𝑦𝐵)))
101, 6, 93imtr4i 291 . . 3 (∃𝑦(𝑥𝑦𝑦 ∈ (𝐴𝐵)) → (𝑥 𝐴𝑥 𝐵))
11 eluni 4908 . . 3 (𝑥 (𝐴𝐵) ↔ ∃𝑦(𝑥𝑦𝑦 ∈ (𝐴𝐵)))
12 elin 3962 . . 3 (𝑥 ∈ ( 𝐴 𝐵) ↔ (𝑥 𝐴𝑥 𝐵))
1310, 11, 123imtr4i 291 . 2 (𝑥 (𝐴𝐵) → 𝑥 ∈ ( 𝐴 𝐵))
1413ssriv 3982 1 (𝐴𝐵) ⊆ ( 𝐴 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wa 394  wex 1774  wcel 2099  cin 3945  wss 3946   cuni 4905
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1537  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-v 3464  df-in 3953  df-ss 3963  df-uni 4906
This theorem is referenced by:  uniinqs  8818  psss  18600  tgval  22946  mapdunirnN  41362
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