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Theorem uniin 3597
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 1522 . . . 4 (∃𝑦((𝑥𝑦𝑦𝐴) ∧ (𝑥𝑦𝑦𝐵)) → (∃𝑦(𝑥𝑦𝑦𝐴) ∧ ∃𝑦(𝑥𝑦𝑦𝐵)))
2 elin 3123 . . . . . . 7 (𝑦 ∈ (𝐴𝐵) ↔ (𝑦𝐴𝑦𝐵))
32anbi2i 430 . . . . . 6 ((𝑥𝑦𝑦 ∈ (𝐴𝐵)) ↔ (𝑥𝑦 ∧ (𝑦𝐴𝑦𝐵)))
4 anandi 524 . . . . . 6 ((𝑥𝑦 ∧ (𝑦𝐴𝑦𝐵)) ↔ ((𝑥𝑦𝑦𝐴) ∧ (𝑥𝑦𝑦𝐵)))
53, 4bitri 173 . . . . 5 ((𝑥𝑦𝑦 ∈ (𝐴𝐵)) ↔ ((𝑥𝑦𝑦𝐴) ∧ (𝑥𝑦𝑦𝐵)))
65exbii 1496 . . . 4 (∃𝑦(𝑥𝑦𝑦 ∈ (𝐴𝐵)) ↔ ∃𝑦((𝑥𝑦𝑦𝐴) ∧ (𝑥𝑦𝑦𝐵)))
7 eluni 3580 . . . . 5 (𝑥 𝐴 ↔ ∃𝑦(𝑥𝑦𝑦𝐴))
8 eluni 3580 . . . . 5 (𝑥 𝐵 ↔ ∃𝑦(𝑥𝑦𝑦𝐵))
97, 8anbi12i 433 . . . 4 ((𝑥 𝐴𝑥 𝐵) ↔ (∃𝑦(𝑥𝑦𝑦𝐴) ∧ ∃𝑦(𝑥𝑦𝑦𝐵)))
101, 6, 93imtr4i 190 . . 3 (∃𝑦(𝑥𝑦𝑦 ∈ (𝐴𝐵)) → (𝑥 𝐴𝑥 𝐵))
11 eluni 3580 . . 3 (𝑥 (𝐴𝐵) ↔ ∃𝑦(𝑥𝑦𝑦 ∈ (𝐴𝐵)))
12 elin 3123 . . 3 (𝑥 ∈ ( 𝐴 𝐵) ↔ (𝑥 𝐴𝑥 𝐵))
1310, 11, 123imtr4i 190 . 2 (𝑥 (𝐴𝐵) → 𝑥 ∈ ( 𝐴 𝐵))
1413ssriv 2946 1 (𝐴𝐵) ⊆ ( 𝐴 𝐵)
Colors of variables: wff set class
Syntax hints:  wa 97  wex 1381  wcel 1393  cin 2913  wss 2914   cuni 3577
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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2556  df-in 2921  df-ss 2928  df-uni 3578
This theorem is referenced by: (None)
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