Theorem tpid1 3830

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 2353	. . 3 ⊢ A = A
2	1	3mix1i 1127	. 2 ⊢ (A = A ∨ A = B ∨ A = C)
3		tpid1.1	. . 3 ⊢ A ∈ V
4	3	eltp 3772	. 2 ⊢ (A ∈ {A, B, C} ↔ (A = A ∨ A = B ∨ A = C))
5	2, 4	mpbir 200	1 ⊢ A ∈ {A, B, C}

Syntax hints:   ∨ w3o 933   = wceq 1642   ∈ wcel 1710  Vcvv 2860  {ctp 3740

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-nin 3212  df-compl 3213  df-un 3215  df-sn 3742  df-pr 3743  df-tp 3744

This theorem is referenced by:  tpnz  3838