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Theorem intpr 3644
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 1370 . . . 4 (∀𝑦((𝑦 = 𝐴𝑥𝑦) ∧ (𝑦 = 𝐵𝑥𝑦)) ↔ (∀𝑦(𝑦 = 𝐴𝑥𝑦) ∧ ∀𝑦(𝑦 = 𝐵𝑥𝑦)))
2 vex 2557 . . . . . . . 8 𝑦 ∈ V
32elpr 3393 . . . . . . 7 (𝑦 ∈ {𝐴, 𝐵} ↔ (𝑦 = 𝐴𝑦 = 𝐵))
43imbi1i 227 . . . . . 6 ((𝑦 ∈ {𝐴, 𝐵} → 𝑥𝑦) ↔ ((𝑦 = 𝐴𝑦 = 𝐵) → 𝑥𝑦))
5 jaob 631 . . . . . 6 (((𝑦 = 𝐴𝑦 = 𝐵) → 𝑥𝑦) ↔ ((𝑦 = 𝐴𝑥𝑦) ∧ (𝑦 = 𝐵𝑥𝑦)))
64, 5bitri 173 . . . . 5 ((𝑦 ∈ {𝐴, 𝐵} → 𝑥𝑦) ↔ ((𝑦 = 𝐴𝑥𝑦) ∧ (𝑦 = 𝐵𝑥𝑦)))
76albii 1359 . . . 4 (∀𝑦(𝑦 ∈ {𝐴, 𝐵} → 𝑥𝑦) ↔ ∀𝑦((𝑦 = 𝐴𝑥𝑦) ∧ (𝑦 = 𝐵𝑥𝑦)))
8 intpr.1 . . . . . 6 𝐴 ∈ V
98clel4 2677 . . . . 5 (𝑥𝐴 ↔ ∀𝑦(𝑦 = 𝐴𝑥𝑦))
10 intpr.2 . . . . . 6 𝐵 ∈ V
1110clel4 2677 . . . . 5 (𝑥𝐵 ↔ ∀𝑦(𝑦 = 𝐵𝑥𝑦))
129, 11anbi12i 433 . . . 4 ((𝑥𝐴𝑥𝐵) ↔ (∀𝑦(𝑦 = 𝐴𝑥𝑦) ∧ ∀𝑦(𝑦 = 𝐵𝑥𝑦)))
131, 7, 123bitr4i 201 . . 3 (∀𝑦(𝑦 ∈ {𝐴, 𝐵} → 𝑥𝑦) ↔ (𝑥𝐴𝑥𝐵))
14 vex 2557 . . . 4 𝑥 ∈ V
1514elint 3618 . . 3 (𝑥 {𝐴, 𝐵} ↔ ∀𝑦(𝑦 ∈ {𝐴, 𝐵} → 𝑥𝑦))
16 elin 3123 . . 3 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
1713, 15, 163bitr4i 201 . 2 (𝑥 {𝐴, 𝐵} ↔ 𝑥 ∈ (𝐴𝐵))
1817eqriv 2037 1 {𝐴, 𝐵} = (𝐴𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 97  wo 629  wal 1241   = wceq 1243  wcel 1393  Vcvv 2554  cin 2913  {cpr 3373   cint 3612
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2556  df-un 2919  df-in 2921  df-sn 3378  df-pr 3379  df-int 3613
This theorem is referenced by:  intprg  3645  op1stb  4205  onintexmid  4291
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