Theorem dftp2 3773

Description: Alternate definition of unordered triple of classes. Special case of Definition 5.3 of [TakeutiZaring] p. 16. (Contributed by NM, 8-Apr-1994.)

Assertion

Ref	Expression
dftp2	⊢ {A, B, C} = {x ∣ (x = A ∨ x = B ∨ x = C)}

Distinct variable groups:   x,A   x,B   x,C

Proof of Theorem dftp2

Step	Hyp	Ref	Expression
1		vex 2863	. . 3 ⊢ x ∈ V
2	1	eltp 3772	. 2 ⊢ (x ∈ {A, B, C} ↔ (x = A ∨ x = B ∨ x = C))
3	2	abbi2i 2465	1 ⊢ {A, B, C} = {x ∣ (x = A ∨ x = B ∨ x = C)}

Syntax hints:   ∨ w3o 933   = wceq 1642  {cab 2339  {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:  tprot  3816  tpid3g  3832