Theorem tpid3 3710

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 2177	. . 3 ⊢ 𝐶 = 𝐶
2	1	3mix3i 1171	. 2 ⊢ (𝐶 = 𝐴 ∨ 𝐶 = 𝐵 ∨ 𝐶 = 𝐶)
3		tpid3.1	. . 3 ⊢ 𝐶 ∈ V
4	3	eltp 3642	. 2 ⊢ (𝐶 ∈ {𝐴, 𝐵, 𝐶} ↔ (𝐶 = 𝐴 ∨ 𝐶 = 𝐵 ∨ 𝐶 = 𝐶))
5	2, 4	mpbir 146	1 ⊢ 𝐶 ∈ {𝐴, 𝐵, 𝐶}

Syntax hints:   ∨ w3o 977   = wceq 1353   ∈ wcel 2148  Vcvv 2739  {ctp 3596

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-3or 979  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2741  df-un 3135  df-sn 3600  df-pr 3601  df-tp 3602

This theorem is referenced by: (None)