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Theorem eltp 3771
 Description: A member of an unordered triple of classes is one of them. Special case of Exercise 1 of [TakeutiZaring] p. 17. (Contributed by NM, 8-Apr-1994.) (Revised by Mario Carneiro, 11-Feb-2015.)
Hypothesis
Ref Expression
eltp.1 A V
Assertion
Ref Expression
eltp (A {B, C, D} ↔ (A = B A = C A = D))

Proof of Theorem eltp
StepHypRef Expression
1 eltp.1 . 2 A V
2 eltpg 3769 . 2 (A V → (A {B, C, D} ↔ (A = B A = C A = D)))
31, 2ax-mp 8 1 (A {B, C, D} ↔ (A = B A = C A = D))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176   ∨ w3o 933   = wceq 1642   ∈ wcel 1710  Vcvv 2859  {ctp 3739 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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:  dftp2  3772  tpid1  3829  tpid2  3830  tpid3  3832
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