ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  intpr GIF version

Theorem intpr 3769
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 1440 . . . 4 (∀𝑦((𝑦 = 𝐴𝑥𝑦) ∧ (𝑦 = 𝐵𝑥𝑦)) ↔ (∀𝑦(𝑦 = 𝐴𝑥𝑦) ∧ ∀𝑦(𝑦 = 𝐵𝑥𝑦)))
2 vex 2660 . . . . . . . 8 𝑦 ∈ V
32elpr 3514 . . . . . . 7 (𝑦 ∈ {𝐴, 𝐵} ↔ (𝑦 = 𝐴𝑦 = 𝐵))
43imbi1i 237 . . . . . 6 ((𝑦 ∈ {𝐴, 𝐵} → 𝑥𝑦) ↔ ((𝑦 = 𝐴𝑦 = 𝐵) → 𝑥𝑦))
5 jaob 682 . . . . . 6 (((𝑦 = 𝐴𝑦 = 𝐵) → 𝑥𝑦) ↔ ((𝑦 = 𝐴𝑥𝑦) ∧ (𝑦 = 𝐵𝑥𝑦)))
64, 5bitri 183 . . . . 5 ((𝑦 ∈ {𝐴, 𝐵} → 𝑥𝑦) ↔ ((𝑦 = 𝐴𝑥𝑦) ∧ (𝑦 = 𝐵𝑥𝑦)))
76albii 1429 . . . 4 (∀𝑦(𝑦 ∈ {𝐴, 𝐵} → 𝑥𝑦) ↔ ∀𝑦((𝑦 = 𝐴𝑥𝑦) ∧ (𝑦 = 𝐵𝑥𝑦)))
8 intpr.1 . . . . . 6 𝐴 ∈ V
98clel4 2791 . . . . 5 (𝑥𝐴 ↔ ∀𝑦(𝑦 = 𝐴𝑥𝑦))
10 intpr.2 . . . . . 6 𝐵 ∈ V
1110clel4 2791 . . . . 5 (𝑥𝐵 ↔ ∀𝑦(𝑦 = 𝐵𝑥𝑦))
129, 11anbi12i 453 . . . 4 ((𝑥𝐴𝑥𝐵) ↔ (∀𝑦(𝑦 = 𝐴𝑥𝑦) ∧ ∀𝑦(𝑦 = 𝐵𝑥𝑦)))
131, 7, 123bitr4i 211 . . 3 (∀𝑦(𝑦 ∈ {𝐴, 𝐵} → 𝑥𝑦) ↔ (𝑥𝐴𝑥𝐵))
14 vex 2660 . . . 4 𝑥 ∈ V
1514elint 3743 . . 3 (𝑥 {𝐴, 𝐵} ↔ ∀𝑦(𝑦 ∈ {𝐴, 𝐵} → 𝑥𝑦))
16 elin 3225 . . 3 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
1713, 15, 163bitr4i 211 . 2 (𝑥 {𝐴, 𝐵} ↔ 𝑥 ∈ (𝐴𝐵))
1817eqriv 2112 1 {𝐴, 𝐵} = (𝐴𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wo 680  wal 1312   = wceq 1314  wcel 1463  Vcvv 2657  cin 3036  {cpr 3494   cint 3737
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-v 2659  df-un 3041  df-in 3043  df-sn 3499  df-pr 3500  df-int 3738
This theorem is referenced by:  intprg  3770  op1stb  4359  onintexmid  4447
  Copyright terms: Public domain W3C validator