Theorem tpid3 3738

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 2196	. . 3 ⊢ 𝐶 = 𝐶
2	1	3mix3i 1173	. 2 ⊢ (𝐶 = 𝐴 ∨ 𝐶 = 𝐵 ∨ 𝐶 = 𝐶)
3		tpid3.1	. . 3 ⊢ 𝐶 ∈ V
4	3	eltp 3670	. 2 ⊢ (𝐶 ∈ {𝐴, 𝐵, 𝐶} ↔ (𝐶 = 𝐴 ∨ 𝐶 = 𝐵 ∨ 𝐶 = 𝐶))
5	2, 4	mpbir 146	1 ⊢ 𝐶 ∈ {𝐴, 𝐵, 𝐶}

Syntax hints:   ∨ w3o 979   = wceq 1364   ∈ wcel 2167  Vcvv 2763  {ctp 3624

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-3or 981  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-un 3161  df-sn 3628  df-pr 3629  df-tp 3630

This theorem is referenced by: (None)