Theorem tpass 4716

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 4592	. 2 ⊢ {𝐵, 𝐶, 𝐴} = ({𝐵, 𝐶} ∪ {𝐴})
2		tprot 4713	. 2 ⊢ {𝐴, 𝐵, 𝐶} = {𝐵, 𝐶, 𝐴}
3		uncom 4108	. 2 ⊢ ({𝐴} ∪ {𝐵, 𝐶}) = ({𝐵, 𝐶} ∪ {𝐴})
4	1, 2, 3	3eqtr4i 2795	1 ⊢ {𝐴, 𝐵, 𝐶} = ({𝐴} ∪ {𝐵, 𝐶})

This proof depends on syntax axioms:   = wceq 1570   ∪ cun 3900  {csn 4587  {cpr 4589  {ctp 4591

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-v 3455  df-un 3907  df-sn 4588  df-pr 4590  df-tp 4592

This theorem is used by:  qdassr  4718  en3  9255  wuntp  10724  symgvalstruct  19530  ex-pw  30917