Theorem tpid1 4713

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

Step	Hyp	Ref	Expression
1		eqid 2737	. . 3 ⊢ 𝐴 = 𝐴
2	1	3mix1i 1335	. 2 ⊢ (𝐴 = 𝐴 ∨ 𝐴 = 𝐵 ∨ 𝐴 = 𝐶)
3		tpid1.1	. . 3 ⊢ 𝐴 ∈ V
4	3	eltp 4634	. 2 ⊢ (𝐴 ∈ {𝐴, 𝐵, 𝐶} ↔ (𝐴 = 𝐴 ∨ 𝐴 = 𝐵 ∨ 𝐴 = 𝐶))
5	2, 4	mpbir 231	1 ⊢ 𝐴 ∈ {𝐴, 𝐵, 𝐶}

Syntax hints:   ∨ 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:  tpnz  4724  hash3tpb  14448  wrdl3s3  14915  cffldtocusgr  29530  cffldtocusgrOLD  29531  usgrwwlks2on  30041  umgrwwlks2on  30042  sgncl  32919  s3rnOLD  33021  cyc3evpm  33226  sgnsf  33238  prodfzo03  34763  circlevma  34802  circlemethhgt  34803  hgt750lemg  34814  hgt750lemb  34816  hgt750lema  34817  hgt750leme  34818  tgoldbachgtde  34820  tgoldbachgt  34823  kur14lem7  35410  kur14lem9  35412  brtpid1  35919  rabren3dioph  43261  fourierdlem102  46654  fourierdlem114  46666  etransclem48  46728  usgrexmpl1tri  48513  usgrexmpl2nb0  48519  usgrexmpl2nb1  48520  usgrexmpl2nb2  48521  usgrexmpl2nb3  48522  usgrexmpl2nb4  48523  usgrexmpl2nb5  48524  gpg3kgrtriex  48577