Theorem tpid2 4448

Description: One of the three elements of an unordered triple. (Contributed by NM, 7-Apr-1994.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)

Hypothesis

Ref	Expression
tpid2.1	⊢ 𝐵 ∈ V

Assertion

Ref	Expression
tpid2	⊢ 𝐵 ∈ {𝐴, 𝐵, 𝐶}

Proof of Theorem tpid2

Step	Hyp	Ref	Expression
1		eqid 2760	. . 3 ⊢ 𝐵 = 𝐵
2	1	3mix2i 1419	. 2 ⊢ (𝐵 = 𝐴 ∨ 𝐵 = 𝐵 ∨ 𝐵 = 𝐶)
3		tpid2.1	. . 3 ⊢ 𝐵 ∈ V
4	3	eltp 4374	. 2 ⊢ (𝐵 ∈ {𝐴, 𝐵, 𝐶} ↔ (𝐵 = 𝐴 ∨ 𝐵 = 𝐵 ∨ 𝐵 = 𝐶))
5	2, 4	mpbir 221	1 ⊢ 𝐵 ∈ {𝐴, 𝐵, 𝐶}

Syntax hints:   ∨ w3o 1071   = wceq 1632   ∈ wcel 2139  Vcvv 3340  {ctp 4325

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-v 3342  df-un 3720  df-sn 4322  df-pr 4324  df-tp 4326

This theorem is referenced by:  wrdl3s3  13906  wwlks2onv  27073  elwwlks2ons3im  27074  elwwlks2ons3OLD  27076  umgrwwlks2on  27078  sgnsf  30038  sgncl  30909  signsw0glem  30939  signsw0g  30942  signswmnd  30943  signswrid  30944  prodfzo03  30990  circlevma  31029  circlemethhgt  31030  hgt750lemg  31041  hgt750lemb  31043  hgt750lema  31044  hgt750leme  31045  tgoldbachgtde  31047  tgoldbachgt  31050  kur14lem7  31501  brtpid2  31910  rabren3dioph  37881  fourierdlem102  40928  fourierdlem114  40940  etransclem48  41002