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Theorem intpr 4709
Description: The intersection of a pair is the intersection of its members. Theorem 71 of [Suppes] p. 42. (Contributed by NM, 14-Oct-1999.)
Hypotheses
Ref Expression
intpr.1 𝐴 ∈ V
intpr.2 𝐵 ∈ V
Assertion
Ref Expression
intpr {𝐴, 𝐵} = (𝐴𝐵)

Proof of Theorem intpr
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 19.26 1959 . . . 4 (∀𝑦((𝑦 = 𝐴𝑥𝑦) ∧ (𝑦 = 𝐵𝑥𝑦)) ↔ (∀𝑦(𝑦 = 𝐴𝑥𝑦) ∧ ∀𝑦(𝑦 = 𝐵𝑥𝑦)))
2 vex 3401 . . . . . . . 8 𝑦 ∈ V
32elpr 4400 . . . . . . 7 (𝑦 ∈ {𝐴, 𝐵} ↔ (𝑦 = 𝐴𝑦 = 𝐵))
43imbi1i 340 . . . . . 6 ((𝑦 ∈ {𝐴, 𝐵} → 𝑥𝑦) ↔ ((𝑦 = 𝐴𝑦 = 𝐵) → 𝑥𝑦))
5 jaob 975 . . . . . 6 (((𝑦 = 𝐴𝑦 = 𝐵) → 𝑥𝑦) ↔ ((𝑦 = 𝐴𝑥𝑦) ∧ (𝑦 = 𝐵𝑥𝑦)))
64, 5bitri 266 . . . . 5 ((𝑦 ∈ {𝐴, 𝐵} → 𝑥𝑦) ↔ ((𝑦 = 𝐴𝑥𝑦) ∧ (𝑦 = 𝐵𝑥𝑦)))
76albii 1904 . . . 4 (∀𝑦(𝑦 ∈ {𝐴, 𝐵} → 𝑥𝑦) ↔ ∀𝑦((𝑦 = 𝐴𝑥𝑦) ∧ (𝑦 = 𝐵𝑥𝑦)))
8 intpr.1 . . . . . 6 𝐴 ∈ V
98clel4 3544 . . . . 5 (𝑥𝐴 ↔ ∀𝑦(𝑦 = 𝐴𝑥𝑦))
10 intpr.2 . . . . . 6 𝐵 ∈ V
1110clel4 3544 . . . . 5 (𝑥𝐵 ↔ ∀𝑦(𝑦 = 𝐵𝑥𝑦))
129, 11anbi12i 614 . . . 4 ((𝑥𝐴𝑥𝐵) ↔ (∀𝑦(𝑦 = 𝐴𝑥𝑦) ∧ ∀𝑦(𝑦 = 𝐵𝑥𝑦)))
131, 7, 123bitr4i 294 . . 3 (∀𝑦(𝑦 ∈ {𝐴, 𝐵} → 𝑥𝑦) ↔ (𝑥𝐴𝑥𝐵))
14 vex 3401 . . . 4 𝑥 ∈ V
1514elint 4682 . . 3 (𝑥 {𝐴, 𝐵} ↔ ∀𝑦(𝑦 ∈ {𝐴, 𝐵} → 𝑥𝑦))
16 elin 4002 . . 3 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
1713, 15, 163bitr4i 294 . 2 (𝑥 {𝐴, 𝐵} ↔ 𝑥 ∈ (𝐴𝐵))
1817eqriv 2810 1 {𝐴, 𝐵} = (𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wo 865  wal 1635   = wceq 1637  wcel 2157  Vcvv 3398  cin 3775  {cpr 4379   cint 4676
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2791
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062  df-clab 2800  df-cleq 2806  df-clel 2809  df-nfc 2944  df-v 3400  df-un 3781  df-in 3783  df-sn 4378  df-pr 4380  df-int 4677
This theorem is referenced by:  intprg  4710  uniintsn  4713  op1stb  5136  fiint  8479  shincli  28555  chincli  28653
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