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Theorem tpid1 4447
 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
tpid1.1 𝐴 ∈ V
Assertion
Ref Expression
tpid1 𝐴 ∈ {𝐴, 𝐵, 𝐶}

Proof of Theorem tpid1
StepHypRef Expression
1 eqid 2760 . . 3 𝐴 = 𝐴
213mix1i 1418 . 2 (𝐴 = 𝐴𝐴 = 𝐵𝐴 = 𝐶)
3 tpid1.1 . . 3 𝐴 ∈ V
43eltp 4374 . 2 (𝐴 ∈ {𝐴, 𝐵, 𝐶} ↔ (𝐴 = 𝐴𝐴 = 𝐵𝐴 = 𝐶))
52, 4mpbir 221 1 𝐴 ∈ {𝐴, 𝐵, 𝐶}
 Colors of variables: wff setvar class 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:  tpnz  4456  wrdl3s3  13926  cffldtocusgr  26574  umgrwwlks2on  27099  sgnsf  30059  sgncl  30930  prodfzo03  31011  circlevma  31050  circlemethhgt  31051  hgt750lemg  31062  hgt750lemb  31064  hgt750lema  31065  hgt750leme  31066  tgoldbachgtde  31068  tgoldbachgt  31071  kur14lem7  31522  kur14lem9  31524  brtpid1  31930  rabren3dioph  37899  fourierdlem102  40946  fourierdlem114  40958  etransclem48  41020
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