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Theorem intpr 3672
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 1384 . . . 4 (∀𝑦((𝑦 = 𝐴𝑥𝑦) ∧ (𝑦 = 𝐵𝑥𝑦)) ↔ (∀𝑦(𝑦 = 𝐴𝑥𝑦) ∧ ∀𝑦(𝑦 = 𝐵𝑥𝑦)))
2 vex 2575 . . . . . . . 8 𝑦 ∈ V
32elpr 3421 . . . . . . 7 (𝑦 ∈ {𝐴, 𝐵} ↔ (𝑦 = 𝐴𝑦 = 𝐵))
43imbi1i 231 . . . . . 6 ((𝑦 ∈ {𝐴, 𝐵} → 𝑥𝑦) ↔ ((𝑦 = 𝐴𝑦 = 𝐵) → 𝑥𝑦))
5 jaob 639 . . . . . 6 (((𝑦 = 𝐴𝑦 = 𝐵) → 𝑥𝑦) ↔ ((𝑦 = 𝐴𝑥𝑦) ∧ (𝑦 = 𝐵𝑥𝑦)))
64, 5bitri 177 . . . . 5 ((𝑦 ∈ {𝐴, 𝐵} → 𝑥𝑦) ↔ ((𝑦 = 𝐴𝑥𝑦) ∧ (𝑦 = 𝐵𝑥𝑦)))
76albii 1373 . . . 4 (∀𝑦(𝑦 ∈ {𝐴, 𝐵} → 𝑥𝑦) ↔ ∀𝑦((𝑦 = 𝐴𝑥𝑦) ∧ (𝑦 = 𝐵𝑥𝑦)))
8 intpr.1 . . . . . 6 𝐴 ∈ V
98clel4 2700 . . . . 5 (𝑥𝐴 ↔ ∀𝑦(𝑦 = 𝐴𝑥𝑦))
10 intpr.2 . . . . . 6 𝐵 ∈ V
1110clel4 2700 . . . . 5 (𝑥𝐵 ↔ ∀𝑦(𝑦 = 𝐵𝑥𝑦))
129, 11anbi12i 441 . . . 4 ((𝑥𝐴𝑥𝐵) ↔ (∀𝑦(𝑦 = 𝐴𝑥𝑦) ∧ ∀𝑦(𝑦 = 𝐵𝑥𝑦)))
131, 7, 123bitr4i 205 . . 3 (∀𝑦(𝑦 ∈ {𝐴, 𝐵} → 𝑥𝑦) ↔ (𝑥𝐴𝑥𝐵))
14 vex 2575 . . . 4 𝑥 ∈ V
1514elint 3646 . . 3 (𝑥 {𝐴, 𝐵} ↔ ∀𝑦(𝑦 ∈ {𝐴, 𝐵} → 𝑥𝑦))
16 elin 3151 . . 3 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
1713, 15, 163bitr4i 205 . 2 (𝑥 {𝐴, 𝐵} ↔ 𝑥 ∈ (𝐴𝐵))
1817eqriv 2051 1 {𝐴, 𝐵} = (𝐴𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wo 637  wal 1255   = wceq 1257  wcel 1407  Vcvv 2572  cin 2941  {cpr 3401   cint 3640
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 638  ax-5 1350  ax-7 1351  ax-gen 1352  ax-ie1 1396  ax-ie2 1397  ax-8 1409  ax-10 1410  ax-11 1411  ax-i12 1412  ax-bndl 1413  ax-4 1414  ax-17 1433  ax-i9 1437  ax-ial 1441  ax-i5r 1442  ax-ext 2036
This theorem depends on definitions:  df-bi 114  df-tru 1260  df-nf 1364  df-sb 1660  df-clab 2041  df-cleq 2047  df-clel 2050  df-nfc 2181  df-v 2574  df-un 2947  df-in 2949  df-sn 3406  df-pr 3407  df-int 3641
This theorem is referenced by:  intprg  3673  op1stb  4234  onintexmid  4322
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