Theorem tpid3 3509

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 2054	. . 3 ⊢ 𝐶 = 𝐶
2	1	3mix3i 1087	. 2 ⊢ (𝐶 = 𝐴 ∨ 𝐶 = 𝐵 ∨ 𝐶 = 𝐶)
3		tpid3.1	. . 3 ⊢ 𝐶 ∈ V
4	3	eltp 3443	. 2 ⊢ (𝐶 ∈ {𝐴, 𝐵, 𝐶} ↔ (𝐶 = 𝐴 ∨ 𝐶 = 𝐵 ∨ 𝐶 = 𝐶))
5	2, 4	mpbir 138	1 ⊢ 𝐶 ∈ {𝐴, 𝐵, 𝐶}

Syntax hints:   ∨ w3o 893   = wceq 1257   ∈ wcel 1407  Vcvv 2572  {ctp 3402

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 638  ax-5 1350  ax-7 1351  ax-gen 1352  ax-ie1 1396  ax-ie2 1397  ax-8 1409  ax-10 1410  ax-11 1411  ax-i12 1412  ax-bndl 1413  ax-4 1414  ax-17 1433  ax-i9 1437  ax-ial 1441  ax-i5r 1442  ax-ext 2036

This theorem depends on definitions:  df-bi 114  df-3or 895  df-tru 1260  df-nf 1364  df-sb 1660  df-clab 2041  df-cleq 2047  df-clel 2050  df-nfc 2181  df-v 2574  df-un 2947  df-sn 3406  df-pr 3407  df-tp 3408

This theorem is referenced by: (None)