Theorem tpass 4718

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 4596	. 2 ⊢ {𝐵, 𝐶, 𝐴} = ({𝐵, 𝐶} ∪ {𝐴})
2		tprot 4715	. 2 ⊢ {𝐴, 𝐵, 𝐶} = {𝐵, 𝐶, 𝐴}
3		uncom 4123	. 2 ⊢ ({𝐴} ∪ {𝐵, 𝐶}) = ({𝐵, 𝐶} ∪ {𝐴})
4	1, 2, 3	3eqtr4i 2763	1 ⊢ {𝐴, 𝐵, 𝐶} = ({𝐴} ∪ {𝐵, 𝐶})

Syntax hints:   = wceq 1540   ∪ cun 3914  {csn 4591  {cpr 4593  {ctp 4595

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-v 3452  df-un 3921  df-sn 4592  df-pr 4594  df-tp 4596

This theorem is referenced by:  qdassr  4720  en3  9233  wuntp  10670  symgvalstruct  19333  ex-pw  30364