Theorem eltp 4712

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 4709	. 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 206   ∨ w3o 1086   = wceq 1537   ∈ wcel 2108  Vcvv 3488  {ctp 4652

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-tru 1540  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-v 3490  df-un 3981  df-sn 4649  df-pr 4651  df-tp 4653

This theorem is referenced by:  dftp2  4714  tpid1  4793  tpid2  4795  brtp  5542  tpres  7238  fntpb  7246  bpoly3  16106  cnfldfun  21401  cnfldfunOLD  21414  gausslemma2dlem0i  27426  2lgsoddprm  27478  sltsolem1  27738  nb3grprlem1  29415  frgr3vlem1  30305  frgr3vlem2  30306  prodtp  32831  s3f1  32913  hgt750lemb  34633  fmtno4prmfac  47446  usgrexmpl2nb0  47846  usgrexmpl2nb3  47849  usgrexmpl2trifr  47852