Theorem eltp 4624

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

Step	Hyp	Ref	Expression
1		eltp.1	. 2 ⊢ 𝐴 ∈ V
2		eltpg 4621	. 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ {𝐵, 𝐶, 𝐷} ↔ (𝐴 = 𝐵 ∨ 𝐴 = 𝐶 ∨ 𝐴 = 𝐷)))
3	1, 2	ax-mp 5	1 ⊢ (𝐴 ∈ {𝐵, 𝐶, 𝐷} ↔ (𝐴 = 𝐵 ∨ 𝐴 = 𝐶 ∨ 𝐴 = 𝐷))

Syntax hints:   ↔ wb 205   ∨ w3o 1085   = wceq 1539   ∈ wcel 2106  Vcvv 3432  {ctp 4565

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-v 3434  df-un 3892  df-sn 4562  df-pr 4564  df-tp 4566

This theorem is referenced by:  dftp2  4625  tpid1  4704  tpid2  4706  tpres  7076  fntpb  7085  bpoly3  15768  cnfldfun  20609  gausslemma2dlem0i  26512  2lgsoddprm  26564  nb3grprlem1  27747  frgr3vlem1  28637  frgr3vlem2  28638  prodtp  31141  s3f1  31221  hgt750lemb  32636  brtp  33717  sltsolem1  33878  fmtno4prmfac  45024