MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  intun Structured version   Visualization version   GIF version

Theorem intun 4661
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 1947 . . . 4 (∀𝑦((𝑦𝐴𝑥𝑦) ∧ (𝑦𝐵𝑥𝑦)) ↔ (∀𝑦(𝑦𝐴𝑥𝑦) ∧ ∀𝑦(𝑦𝐵𝑥𝑦)))
2 elun 3896 . . . . . . 7 (𝑦 ∈ (𝐴𝐵) ↔ (𝑦𝐴𝑦𝐵))
32imbi1i 338 . . . . . 6 ((𝑦 ∈ (𝐴𝐵) → 𝑥𝑦) ↔ ((𝑦𝐴𝑦𝐵) → 𝑥𝑦))
4 jaob 857 . . . . . 6 (((𝑦𝐴𝑦𝐵) → 𝑥𝑦) ↔ ((𝑦𝐴𝑥𝑦) ∧ (𝑦𝐵𝑥𝑦)))
53, 4bitri 264 . . . . 5 ((𝑦 ∈ (𝐴𝐵) → 𝑥𝑦) ↔ ((𝑦𝐴𝑥𝑦) ∧ (𝑦𝐵𝑥𝑦)))
65albii 1896 . . . 4 (∀𝑦(𝑦 ∈ (𝐴𝐵) → 𝑥𝑦) ↔ ∀𝑦((𝑦𝐴𝑥𝑦) ∧ (𝑦𝐵𝑥𝑦)))
7 vex 3343 . . . . . 6 𝑥 ∈ V
87elint 4633 . . . . 5 (𝑥 𝐴 ↔ ∀𝑦(𝑦𝐴𝑥𝑦))
97elint 4633 . . . . 5 (𝑥 𝐵 ↔ ∀𝑦(𝑦𝐵𝑥𝑦))
108, 9anbi12i 735 . . . 4 ((𝑥 𝐴𝑥 𝐵) ↔ (∀𝑦(𝑦𝐴𝑥𝑦) ∧ ∀𝑦(𝑦𝐵𝑥𝑦)))
111, 6, 103bitr4i 292 . . 3 (∀𝑦(𝑦 ∈ (𝐴𝐵) → 𝑥𝑦) ↔ (𝑥 𝐴𝑥 𝐵))
127elint 4633 . . 3 (𝑥 (𝐴𝐵) ↔ ∀𝑦(𝑦 ∈ (𝐴𝐵) → 𝑥𝑦))
13 elin 3939 . . 3 (𝑥 ∈ ( 𝐴 𝐵) ↔ (𝑥 𝐴𝑥 𝐵))
1411, 12, 133bitr4i 292 . 2 (𝑥 (𝐴𝐵) ↔ 𝑥 ∈ ( 𝐴 𝐵))
1514eqriv 2757 1 (𝐴𝐵) = ( 𝐴 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 382  wa 383  wal 1630   = wceq 1632  wcel 2139  cun 3713  cin 3714   cint 4627
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-v 3342  df-un 3720  df-in 3722  df-int 4628
This theorem is referenced by:  intunsn  4668  riinint  5537  fiin  8493  elfiun  8501  elrfi  37759
  Copyright terms: Public domain W3C validator