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Theorem uniin 4650
Description: The class union of the intersection of two classes. Exercise 4.12(n) of [Mendelson] p. 235. See uniinqs 8065 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 1968 . . . 4 (∃𝑦((𝑥𝑦𝑦𝐴) ∧ (𝑥𝑦𝑦𝐵)) → (∃𝑦(𝑥𝑦𝑦𝐴) ∧ ∃𝑦(𝑥𝑦𝑦𝐵)))
2 elin 3994 . . . . . . 7 (𝑦 ∈ (𝐴𝐵) ↔ (𝑦𝐴𝑦𝐵))
32anbi2i 617 . . . . . 6 ((𝑥𝑦𝑦 ∈ (𝐴𝐵)) ↔ (𝑥𝑦 ∧ (𝑦𝐴𝑦𝐵)))
4 anandi 667 . . . . . 6 ((𝑥𝑦 ∧ (𝑦𝐴𝑦𝐵)) ↔ ((𝑥𝑦𝑦𝐴) ∧ (𝑥𝑦𝑦𝐵)))
53, 4bitri 267 . . . . 5 ((𝑥𝑦𝑦 ∈ (𝐴𝐵)) ↔ ((𝑥𝑦𝑦𝐴) ∧ (𝑥𝑦𝑦𝐵)))
65exbii 1944 . . . 4 (∃𝑦(𝑥𝑦𝑦 ∈ (𝐴𝐵)) ↔ ∃𝑦((𝑥𝑦𝑦𝐴) ∧ (𝑥𝑦𝑦𝐵)))
7 eluni 4631 . . . . 5 (𝑥 𝐴 ↔ ∃𝑦(𝑥𝑦𝑦𝐴))
8 eluni 4631 . . . . 5 (𝑥 𝐵 ↔ ∃𝑦(𝑥𝑦𝑦𝐵))
97, 8anbi12i 621 . . . 4 ((𝑥 𝐴𝑥 𝐵) ↔ (∃𝑦(𝑥𝑦𝑦𝐴) ∧ ∃𝑦(𝑥𝑦𝑦𝐵)))
101, 6, 93imtr4i 284 . . 3 (∃𝑦(𝑥𝑦𝑦 ∈ (𝐴𝐵)) → (𝑥 𝐴𝑥 𝐵))
11 eluni 4631 . . 3 (𝑥 (𝐴𝐵) ↔ ∃𝑦(𝑥𝑦𝑦 ∈ (𝐴𝐵)))
12 elin 3994 . . 3 (𝑥 ∈ ( 𝐴 𝐵) ↔ (𝑥 𝐴𝑥 𝐵))
1310, 11, 123imtr4i 284 . 2 (𝑥 (𝐴𝐵) → 𝑥 ∈ ( 𝐴 𝐵))
1413ssriv 3802 1 (𝐴𝐵) ⊆ ( 𝐴 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wa 385  wex 1875  wcel 2157  cin 3768  wss 3769   cuni 4628
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-ext 2777
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-v 3387  df-in 3776  df-ss 3783  df-uni 4629
This theorem is referenced by:  uniinqs  8065  psss  17529  tgval  21088  mapdunirnN  37671
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