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Theorem intprg 4871
 Description: The intersection of a pair is the intersection of its members. Closed form of intpr 4872. 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 3413 . . . . . 6 𝑥 ∈ V
21elint 4844 . . . . 5 (𝑥 {𝐴, 𝐵} ↔ ∀𝑦(𝑦 ∈ {𝐴, 𝐵} → 𝑥𝑦))
3 vex 3413 . . . . . . . . 9 𝑦 ∈ V
43elpr 4545 . . . . . . . 8 (𝑦 ∈ {𝐴, 𝐵} ↔ (𝑦 = 𝐴𝑦 = 𝐵))
54imbi1i 353 . . . . . . 7 ((𝑦 ∈ {𝐴, 𝐵} → 𝑥𝑦) ↔ ((𝑦 = 𝐴𝑦 = 𝐵) → 𝑥𝑦))
6 jaob 959 . . . . . . 7 (((𝑦 = 𝐴𝑦 = 𝐵) → 𝑥𝑦) ↔ ((𝑦 = 𝐴𝑥𝑦) ∧ (𝑦 = 𝐵𝑥𝑦)))
75, 6bitri 278 . . . . . 6 ((𝑦 ∈ {𝐴, 𝐵} → 𝑥𝑦) ↔ ((𝑦 = 𝐴𝑥𝑦) ∧ (𝑦 = 𝐵𝑥𝑦)))
87albii 1821 . . . . 5 (∀𝑦(𝑦 ∈ {𝐴, 𝐵} → 𝑥𝑦) ↔ ∀𝑦((𝑦 = 𝐴𝑥𝑦) ∧ (𝑦 = 𝐵𝑥𝑦)))
9 19.26 1871 . . . . 5 (∀𝑦((𝑦 = 𝐴𝑥𝑦) ∧ (𝑦 = 𝐵𝑥𝑦)) ↔ (∀𝑦(𝑦 = 𝐴𝑥𝑦) ∧ ∀𝑦(𝑦 = 𝐵𝑥𝑦)))
102, 8, 93bitri 300 . . . 4 (𝑥 {𝐴, 𝐵} ↔ (∀𝑦(𝑦 = 𝐴𝑥𝑦) ∧ ∀𝑦(𝑦 = 𝐵𝑥𝑦)))
11 elin 3874 . . . . 5 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
12 clel4g 3575 . . . . . 6 (𝐴𝑉 → (𝑥𝐴 ↔ ∀𝑦(𝑦 = 𝐴𝑥𝑦)))
13 clel4g 3575 . . . . . 6 (𝐵𝑊 → (𝑥𝐵 ↔ ∀𝑦(𝑦 = 𝐵𝑥𝑦)))
1412, 13bi2anan9 638 . . . . 5 ((𝐴𝑉𝐵𝑊) → ((𝑥𝐴𝑥𝐵) ↔ (∀𝑦(𝑦 = 𝐴𝑥𝑦) ∧ ∀𝑦(𝑦 = 𝐵𝑥𝑦))))
1511, 14syl5rbb 287 . . . 4 ((𝐴𝑉𝐵𝑊) → ((∀𝑦(𝑦 = 𝐴𝑥𝑦) ∧ ∀𝑦(𝑦 = 𝐵𝑥𝑦)) ↔ 𝑥 ∈ (𝐴𝐵)))
1610, 15syl5bb 286 . . 3 ((𝐴𝑉𝐵𝑊) → (𝑥 {𝐴, 𝐵} ↔ 𝑥 ∈ (𝐴𝐵)))
1716alrimiv 1928 . 2 ((𝐴𝑉𝐵𝑊) → ∀𝑥(𝑥 {𝐴, 𝐵} ↔ 𝑥 ∈ (𝐴𝐵)))
18 dfcleq 2751 . 2 ( {𝐴, 𝐵} = (𝐴𝐵) ↔ ∀𝑥(𝑥 {𝐴, 𝐵} ↔ 𝑥 ∈ (𝐴𝐵)))
1917, 18sylibr 237 1 ((𝐴𝑉𝐵𝑊) → {𝐴, 𝐵} = (𝐴𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∨ wo 844  ∀wal 1536   = wceq 1538   ∈ wcel 2111   ∩ cin 3857  {cpr 4524  ∩ cint 4838 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 2113  ax-9 2121  ax-ext 2729 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-v 3411  df-un 3863  df-in 3865  df-sn 4523  df-pr 4525  df-int 4839 This theorem is referenced by:  intpr  4872  intsng  4875  inelfi  8915  mreincl  16928  subrgin  19626  lssincl  19805  incld  21743  difelsiga  31620  inelpisys  31641  bj-prmoore  34810  inidl  35748
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