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Theorem eltp 4433
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 = 𝐵𝐴 = 𝐶𝐴 = 𝐷))

Proof of Theorem eltp
StepHypRef Expression
1 eltp.1 . 2 𝐴 ∈ V
2 eltpg 4430 . 2 (𝐴 ∈ V → (𝐴 ∈ {𝐵, 𝐶, 𝐷} ↔ (𝐴 = 𝐵𝐴 = 𝐶𝐴 = 𝐷)))
31, 2ax-mp 5 1 (𝐴 ∈ {𝐵, 𝐶, 𝐷} ↔ (𝐴 = 𝐵𝐴 = 𝐶𝐴 = 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wb 197  w3o 1099   = wceq 1637  wcel 2157  Vcvv 3402  {ctp 4385
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2795
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062  df-clab 2804  df-cleq 2810  df-clel 2813  df-nfc 2948  df-v 3404  df-un 3785  df-sn 4382  df-pr 4384  df-tp 4386
This theorem is referenced by:  dftp2  4434  tpid1  4505  tpid2  4506  tpres  6700  fntpb  6707  bpoly3  15028  cnfldfunALT  19986  gausslemma2dlem0i  25325  2lgsoddprm  25377  nb3grprlem1  26520  frgr3vlem1  27470  frgr3vlem2  27471  prodtp  29922  hgt750lemb  31081  brtp  31982  sltsolem1  32168  fmtno4prmfac  42076
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