Theorem eltp 4634

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 4631	. 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 1542   ∈ wcel 2114  Vcvv 3430  {ctp 4572

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-v 3432  df-un 3895  df-sn 4569  df-pr 4571  df-tp 4573

This theorem is referenced by:  dftp2  4636  tpid1  4713  tpid2  4715  brtp  5471  tpres  7149  fntpb  7157  bpoly3  16014  cnfldfun  21358  cnfldfunOLD  21371  gausslemma2dlem0i  27341  2lgsoddprm  27393  ltssolem1  27653  nb3grprlem1  29463  frgr3vlem1  30358  frgr3vlem2  30359  prodtp  32915  s3f1  33022  hgt750lemb  34816  fmtno4prmfac  48047  usgrexmpl2nb0  48519  usgrexmpl2nb3  48522  usgrexmpl2trifr  48525  gpgnbgrvtx0  48562  gpgnbgrvtx1  48563