New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  tpid1 GIF version

Theorem tpid1 3829
 Description: One of the three elements of an unordered triple. (Contributed by NM, 7-Apr-1994.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
Hypothesis
Ref Expression
tpid1.1 A V
Assertion
Ref Expression
tpid1 A {A, B, C}

Proof of Theorem tpid1
StepHypRef Expression
1 eqid 2353 . . 3 A = A
213mix1i 1127 . 2 (A = A A = B A = C)
3 tpid1.1 . . 3 A V
43eltp 3771 . 2 (A {A, B, C} ↔ (A = A A = B A = C))
52, 4mpbir 200 1 A {A, B, C}
 Colors of variables: wff setvar class Syntax hints:   ∨ w3o 933   = wceq 1642   ∈ wcel 1710  Vcvv 2859  {ctp 3739 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-nin 3211  df-compl 3212  df-un 3214  df-sn 3741  df-pr 3742  df-tp 3743 This theorem is referenced by:  tpnz  3837
 Copyright terms: Public domain W3C validator