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Theorem uniin 3756
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 1610 . . . 4 (∃𝑦((𝑥𝑦𝑦𝐴) ∧ (𝑥𝑦𝑦𝐵)) → (∃𝑦(𝑥𝑦𝑦𝐴) ∧ ∃𝑦(𝑥𝑦𝑦𝐵)))
2 elin 3259 . . . . . . 7 (𝑦 ∈ (𝐴𝐵) ↔ (𝑦𝐴𝑦𝐵))
32anbi2i 452 . . . . . 6 ((𝑥𝑦𝑦 ∈ (𝐴𝐵)) ↔ (𝑥𝑦 ∧ (𝑦𝐴𝑦𝐵)))
4 anandi 579 . . . . . 6 ((𝑥𝑦 ∧ (𝑦𝐴𝑦𝐵)) ↔ ((𝑥𝑦𝑦𝐴) ∧ (𝑥𝑦𝑦𝐵)))
53, 4bitri 183 . . . . 5 ((𝑥𝑦𝑦 ∈ (𝐴𝐵)) ↔ ((𝑥𝑦𝑦𝐴) ∧ (𝑥𝑦𝑦𝐵)))
65exbii 1584 . . . 4 (∃𝑦(𝑥𝑦𝑦 ∈ (𝐴𝐵)) ↔ ∃𝑦((𝑥𝑦𝑦𝐴) ∧ (𝑥𝑦𝑦𝐵)))
7 eluni 3739 . . . . 5 (𝑥 𝐴 ↔ ∃𝑦(𝑥𝑦𝑦𝐴))
8 eluni 3739 . . . . 5 (𝑥 𝐵 ↔ ∃𝑦(𝑥𝑦𝑦𝐵))
97, 8anbi12i 455 . . . 4 ((𝑥 𝐴𝑥 𝐵) ↔ (∃𝑦(𝑥𝑦𝑦𝐴) ∧ ∃𝑦(𝑥𝑦𝑦𝐵)))
101, 6, 93imtr4i 200 . . 3 (∃𝑦(𝑥𝑦𝑦 ∈ (𝐴𝐵)) → (𝑥 𝐴𝑥 𝐵))
11 eluni 3739 . . 3 (𝑥 (𝐴𝐵) ↔ ∃𝑦(𝑥𝑦𝑦 ∈ (𝐴𝐵)))
12 elin 3259 . . 3 (𝑥 ∈ ( 𝐴 𝐵) ↔ (𝑥 𝐴𝑥 𝐵))
1310, 11, 123imtr4i 200 . 2 (𝑥 (𝐴𝐵) → 𝑥 ∈ ( 𝐴 𝐵))
1413ssriv 3101 1 (𝐴𝐵) ⊆ ( 𝐴 𝐵)
Colors of variables: wff set class
Syntax hints:  wa 103  wex 1468  wcel 1480  cin 3070  wss 3071   cuni 3736
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-in 3077  df-ss 3084  df-uni 3737
This theorem is referenced by:  tgval  12232
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