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Theorem intprg 4986
Description: The intersection of a pair is the intersection of its members. Closed form of intpr 4987. Theorem 71 of [Suppes] p. 42. (Contributed by FL, 27-Apr-2008.) (Proof shortened by BJ, 1-Sep-2024.)
Assertion
Ref Expression
intprg ((𝐴𝑉𝐵𝑊) → {𝐴, 𝐵} = (𝐴𝐵))

Proof of Theorem intprg
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 3479 . . . . . 6 𝑥 ∈ V
21elint 4957 . . . . 5 (𝑥 {𝐴, 𝐵} ↔ ∀𝑦(𝑦 ∈ {𝐴, 𝐵} → 𝑥𝑦))
3 vex 3479 . . . . . . . . 9 𝑦 ∈ V
43elpr 4652 . . . . . . . 8 (𝑦 ∈ {𝐴, 𝐵} ↔ (𝑦 = 𝐴𝑦 = 𝐵))
54imbi1i 350 . . . . . . 7 ((𝑦 ∈ {𝐴, 𝐵} → 𝑥𝑦) ↔ ((𝑦 = 𝐴𝑦 = 𝐵) → 𝑥𝑦))
6 jaob 961 . . . . . . 7 (((𝑦 = 𝐴𝑦 = 𝐵) → 𝑥𝑦) ↔ ((𝑦 = 𝐴𝑥𝑦) ∧ (𝑦 = 𝐵𝑥𝑦)))
75, 6bitri 275 . . . . . 6 ((𝑦 ∈ {𝐴, 𝐵} → 𝑥𝑦) ↔ ((𝑦 = 𝐴𝑥𝑦) ∧ (𝑦 = 𝐵𝑥𝑦)))
87albii 1822 . . . . 5 (∀𝑦(𝑦 ∈ {𝐴, 𝐵} → 𝑥𝑦) ↔ ∀𝑦((𝑦 = 𝐴𝑥𝑦) ∧ (𝑦 = 𝐵𝑥𝑦)))
9 19.26 1874 . . . . 5 (∀𝑦((𝑦 = 𝐴𝑥𝑦) ∧ (𝑦 = 𝐵𝑥𝑦)) ↔ (∀𝑦(𝑦 = 𝐴𝑥𝑦) ∧ ∀𝑦(𝑦 = 𝐵𝑥𝑦)))
102, 8, 93bitri 297 . . . 4 (𝑥 {𝐴, 𝐵} ↔ (∀𝑦(𝑦 = 𝐴𝑥𝑦) ∧ ∀𝑦(𝑦 = 𝐵𝑥𝑦)))
11 elin 3965 . . . . 5 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
12 clel4g 3653 . . . . . 6 (𝐴𝑉 → (𝑥𝐴 ↔ ∀𝑦(𝑦 = 𝐴𝑥𝑦)))
13 clel4g 3653 . . . . . 6 (𝐵𝑊 → (𝑥𝐵 ↔ ∀𝑦(𝑦 = 𝐵𝑥𝑦)))
1412, 13bi2anan9 638 . . . . 5 ((𝐴𝑉𝐵𝑊) → ((𝑥𝐴𝑥𝐵) ↔ (∀𝑦(𝑦 = 𝐴𝑥𝑦) ∧ ∀𝑦(𝑦 = 𝐵𝑥𝑦))))
1511, 14bitr2id 284 . . . 4 ((𝐴𝑉𝐵𝑊) → ((∀𝑦(𝑦 = 𝐴𝑥𝑦) ∧ ∀𝑦(𝑦 = 𝐵𝑥𝑦)) ↔ 𝑥 ∈ (𝐴𝐵)))
1610, 15bitrid 283 . . 3 ((𝐴𝑉𝐵𝑊) → (𝑥 {𝐴, 𝐵} ↔ 𝑥 ∈ (𝐴𝐵)))
1716alrimiv 1931 . 2 ((𝐴𝑉𝐵𝑊) → ∀𝑥(𝑥 {𝐴, 𝐵} ↔ 𝑥 ∈ (𝐴𝐵)))
18 dfcleq 2726 . 2 ( {𝐴, 𝐵} = (𝐴𝐵) ↔ ∀𝑥(𝑥 {𝐴, 𝐵} ↔ 𝑥 ∈ (𝐴𝐵)))
1917, 18sylibr 233 1 ((𝐴𝑉𝐵𝑊) → {𝐴, 𝐵} = (𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  wo 846  wal 1540   = wceq 1542  wcel 2107  cin 3948  {cpr 4631   cint 4951
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3477  df-un 3954  df-in 3956  df-sn 4630  df-pr 4632  df-int 4952
This theorem is referenced by:  intpr  4987  intsng  4990  inelfi  9413  mreincl  17543  subrgin  20343  lssincl  20576  incld  22547  difelsiga  33131  inelpisys  33152  bj-prmoore  35996  inidl  36898  subrngin  46740  toplatmeet  47628
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