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Theorem intun 4873
Description: The class intersection of the union of two classes. Theorem 78 of [Suppes] p. 42. (Contributed by NM, 22-Sep-2002.)
Assertion
Ref Expression
intun (𝐴𝐵) = ( 𝐴 𝐵)

Proof of Theorem intun
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 19.26 1871 . . . 4 (∀𝑦((𝑦𝐴𝑥𝑦) ∧ (𝑦𝐵𝑥𝑦)) ↔ (∀𝑦(𝑦𝐴𝑥𝑦) ∧ ∀𝑦(𝑦𝐵𝑥𝑦)))
2 elunant 4108 . . . . 5 ((𝑦 ∈ (𝐴𝐵) → 𝑥𝑦) ↔ ((𝑦𝐴𝑥𝑦) ∧ (𝑦𝐵𝑥𝑦)))
32albii 1821 . . . 4 (∀𝑦(𝑦 ∈ (𝐴𝐵) → 𝑥𝑦) ↔ ∀𝑦((𝑦𝐴𝑥𝑦) ∧ (𝑦𝐵𝑥𝑦)))
4 vex 3447 . . . . . 6 𝑥 ∈ V
54elint 4847 . . . . 5 (𝑥 𝐴 ↔ ∀𝑦(𝑦𝐴𝑥𝑦))
64elint 4847 . . . . 5 (𝑥 𝐵 ↔ ∀𝑦(𝑦𝐵𝑥𝑦))
75, 6anbi12i 629 . . . 4 ((𝑥 𝐴𝑥 𝐵) ↔ (∀𝑦(𝑦𝐴𝑥𝑦) ∧ ∀𝑦(𝑦𝐵𝑥𝑦)))
81, 3, 73bitr4i 306 . . 3 (∀𝑦(𝑦 ∈ (𝐴𝐵) → 𝑥𝑦) ↔ (𝑥 𝐴𝑥 𝐵))
94elint 4847 . . 3 (𝑥 (𝐴𝐵) ↔ ∀𝑦(𝑦 ∈ (𝐴𝐵) → 𝑥𝑦))
10 elin 3900 . . 3 (𝑥 ∈ ( 𝐴 𝐵) ↔ (𝑥 𝐴𝑥 𝐵))
118, 9, 103bitr4i 306 . 2 (𝑥 (𝐴𝐵) ↔ 𝑥 ∈ ( 𝐴 𝐵))
1211eqriv 2798 1 (𝐴𝐵) = ( 𝐴 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  wal 1536   = wceq 1538  wcel 2112  cun 3882  cin 3883   cint 4841
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-ext 2773
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-v 3446  df-un 3889  df-in 3891  df-int 4842
This theorem is referenced by:  intunsn  4880  riinint  5808  fiin  8874  elfiun  8882  elrfi  39632
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