 Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  dftp2 Structured version   Visualization version   GIF version

Theorem dftp2 4528
 Description: Alternate definition of unordered triple of classes. Special case of Definition 5.3 of [TakeutiZaring] p. 16. (Contributed by NM, 8-Apr-1994.)
Assertion
Ref Expression
dftp2 {𝐴, 𝐵, 𝐶} = {𝑥 ∣ (𝑥 = 𝐴𝑥 = 𝐵𝑥 = 𝐶)}
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐶

Proof of Theorem dftp2
StepHypRef Expression
1 vex 3435 . . 3 𝑥 ∈ V
21eltp 4527 . 2 (𝑥 ∈ {𝐴, 𝐵, 𝐶} ↔ (𝑥 = 𝐴𝑥 = 𝐵𝑥 = 𝐶))
32abbi2i 2920 1 {𝐴, 𝐵, 𝐶} = {𝑥 ∣ (𝑥 = 𝐴𝑥 = 𝐵𝑥 = 𝐶)}
 Colors of variables: wff setvar class Syntax hints:   ∨ w3o 1077   = wceq 1520  {cab 2773  {ctp 4470 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1775  ax-4 1789  ax-5 1886  ax-6 1945  ax-7 1990  ax-8 2081  ax-9 2089  ax-10 2110  ax-11 2124  ax-12 2139  ax-ext 2767 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1079  df-tru 1523  df-ex 1760  df-nf 1764  df-sb 2041  df-clab 2774  df-cleq 2786  df-clel 2861  df-nfc 2933  df-v 3434  df-un 3859  df-sn 4467  df-pr 4469  df-tp 4471 This theorem is referenced by:  tprot  4586  en3lplem2  8911  tpid3gVD  40667  en3lplem2VD  40669
 Copyright terms: Public domain W3C validator