Theorem eltp 4651

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 4648	. 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 209   ∨ w3o 1100   = wceq 1563   ∈ wcel 2145  Vcvv 3457  {ctp 4589

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-tru 1566  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-v 3459  df-un 3912  df-sn 4586  df-pr 4588  df-tp 4590

This theorem is referenced by:  dftp2  4653  tpid1  4730  tpid2  4732  brtp  5498  tpres  7189  fntpb  7197  bpoly3  16102  cnfldfun  21496  gausslemma2dlem0i  27486  2lgsoddprm  27538  ltssolem1  27797  nb3grprlem1  29639  frgr3vlem1  30533  frgr3vlem2  30534  prodtp  33084  s3f1  33180  hgt750lemb  34960  fmtno4prmfac  48179  usgrexmpl2nb0  48651  usgrexmpl2nb3  48654  usgrexmpl2trifr  48657  gpgnbgrvtx0  48694  gpgnbgrvtx1  48695