Theorem tpid3 3506

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
tpid3.1	⊢ 𝐶 ∈ V

Assertion

Ref	Expression
tpid3	⊢ 𝐶 ∈ {𝐴, 𝐵, 𝐶}

Proof of Theorem tpid3

Step	Hyp	Ref	Expression
1		eqid 2081	. . 3 ⊢ 𝐶 = 𝐶
2	1	3mix3i 1112	. 2 ⊢ (𝐶 = 𝐴 ∨ 𝐶 = 𝐵 ∨ 𝐶 = 𝐶)
3		tpid3.1	. . 3 ⊢ 𝐶 ∈ V
4	3	eltp 3440	. 2 ⊢ (𝐶 ∈ {𝐴, 𝐵, 𝐶} ↔ (𝐶 = 𝐴 ∨ 𝐶 = 𝐵 ∨ 𝐶 = 𝐶))
5	2, 4	mpbir 144	1 ⊢ 𝐶 ∈ {𝐴, 𝐵, 𝐶}

Syntax hints:   ∨ w3o 918   = wceq 1284   ∈ wcel 1433  Vcvv 2601  {ctp 3400

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-3or 920  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-un 2977  df-sn 3404  df-pr 3405  df-tp 3406

This theorem is referenced by: (None)