 Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  tpass Structured version   GIF version

Theorem tpass 3440
 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
StepHypRef Expression
1 df-tp 3358 . 2 {B, 𝐶, A} = ({B, 𝐶} ∪ {A})
2 tprot 3437 . 2 {A, B, 𝐶} = {B, 𝐶, A}
3 uncom 3064 . 2 ({A} ∪ {B, 𝐶}) = ({B, 𝐶} ∪ {A})
41, 2, 33eqtr4i 2052 1 {A, B, 𝐶} = ({A} ∪ {B, 𝐶})
 Colors of variables: wff set class Syntax hints:   = wceq 1228   ∪ cun 2892  {csn 3350  {cpr 3351  {ctp 3352 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004 This theorem depends on definitions:  df-bi 110  df-3or 874  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-v 2537  df-un 2899  df-sn 3356  df-pr 3357  df-tp 3358 This theorem is referenced by:  qdassr  3442
 Copyright terms: Public domain W3C validator