Theorem tpid1 4725

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 2760	. . 3 ⊢ 𝐴 = 𝐴
2	1	3mix1i 1352	. 2 ⊢ (𝐴 = 𝐴 ∨ 𝐴 = 𝐵 ∨ 𝐴 = 𝐶)
3		tpid1.1	. . 3 ⊢ 𝐴 ∈ V
4	3	eltp 4646	. 2 ⊢ (𝐴 ∈ {𝐴, 𝐵, 𝐶} ↔ (𝐴 = 𝐴 ∨ 𝐴 = 𝐵 ∨ 𝐴 = 𝐶))
5	2, 4	mpbir 234	1 ⊢ 𝐴 ∈ {𝐴, 𝐵, 𝐶}

This proof depends on syntax axioms:   ∨ w3o 1102   = wceq 1569   ∈ wcel 2144  Vcvv 3450  {ctp 4588

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-tru 1572  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-v 3452  df-un 4151  df-sn 4585  df-pr 4587  df-tp 4589

This theorem is used by:  tpnz  4736  hash3tpb  14643  s3rex  15106  wrdl3s3  15120  sgncl  15255  elcgrabasi  29380  cffldtocusgr  30033  usgrwwlks2on  30552  umgrwwlks2on  30553  cyc3evpm  33757  sgnsf  33769  prodfzo03  35278  circlevma  35317  circlemethhgt  35318  hgt750lemg  35329  hgt750lemb  35331  hgt750lema  35332  hgt750leme  35333  tgoldbachgtde  35335  tgoldbachgt  35338  kur14lem7  36063  kur14lem9  36065  brtpid1  36578  rabren3dioph  43915  fourierdlem102  47311  fourierdlem114  47323  etransclem48  47385  usgrexmpl1tri  49216  usgrexmpl2nb0  49222  usgrexmpl2nb1  49223  usgrexmpl2nb2  49224  usgrexmpl2nb3  49225  usgrexmpl2nb4  49226  usgrexmpl2nb5  49227  gpg3kgrtriex  49280