Theorem tpid1 4739

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 2766	. . 3 ⊢ 𝐴 = 𝐴
2	1	3mix1i 1352	. 2 ⊢ (𝐴 = 𝐴 ∨ 𝐴 = 𝐵 ∨ 𝐴 = 𝐶)
3		tpid1.1	. . 3 ⊢ 𝐴 ∈ V
4	3	eltp 4660	. 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 1570   ∈ wcel 2146  Vcvv 3458  {ctp 4598

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2738

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-v 3460  df-un 3913  df-sn 4595  df-pr 4597  df-tp 4599

This theorem is used by:  tpnz  4750  hash3tpb  14552  wrdl3s3  15025  sgncl  15160  cffldtocusgr  29834  usgrwwlks2on  30344  umgrwwlks2on  30345  cyc3evpm  33501  sgnsf  33513  prodfzo03  35022  circlevma  35061  circlemethhgt  35062  hgt750lemg  35073  hgt750lemb  35075  hgt750lema  35076  hgt750leme  35077  tgoldbachgtde  35079  tgoldbachgt  35082  kur14lem7  35725  kur14lem9  35727  brtpid1  36234  rabren3dioph  43583  fourierdlem102  46963  fourierdlem114  46975  etransclem48  47037  usgrexmpl1tri  48831  usgrexmpl2nb0  48837  usgrexmpl2nb1  48838  usgrexmpl2nb2  48839  usgrexmpl2nb3  48840  usgrexmpl2nb4  48841  usgrexmpl2nb5  48842  gpg3kgrtriex  48895