Theorem tpass 3819

Description: Split off the first element of an unordered triple. (Contributed by Mario Carneiro, 5-Jan-2016.)

Assertion

Ref	Expression
tpass	⊢ {A, B, C} = ({A} ∪ {B, C})

Proof of Theorem tpass

Step	Hyp	Ref	Expression
1		df-tp 3744	. 2 ⊢ {B, C, A} = ({B, C} ∪ {A})
2		tprot 3816	. 2 ⊢ {A, B, C} = {B, C, A}
3		uncom 3409	. 2 ⊢ ({A} ∪ {B, C}) = ({B, C} ∪ {A})
4	1, 2, 3	3eqtr4i 2383	1 ⊢ {A, B, C} = ({A} ∪ {B, C})

Syntax hints:   = wceq 1642   ∪ cun 3208  {csn 3738  {cpr 3739  {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:  qdassr  3821