Theorem tpi2 2456

Description: One of the three elements of an unordered triple.

Hypothesis

Ref	Expression
tpi2.1	|- B e. V

Assertion

Ref	Expression
tpi2	|- B e. {A, B, C}

Proof of Theorem tpi2

Step	Hyp	Ref	Expression
1		tpi2.1	. . 3 |- B e. V
2	1	pri2 2451	. 2 |- B e. {A, B}
3		elun1 2197	. . 3 |- (B e. {A, B} -> B e. ({A, B} u. {C}))
4		df-tp 2415	. . 3 |- {A, B, C} = ({A, B} u. {C})
5	3, 4	syl6eleqr 1559	. 2 |- (B e. {A, B} -> B e. {A, B, C})
6	2, 5	ax-mp 7	1 |- B e. {A, B, C}

Syntax hints:   e. wcel 958  Vcvv 1811   u. cun 2045  {csn 2409  {cpr 2410  {ctp 2414

This theorem is referenced by:  tpss 2476  fr3nr 2926

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-v 1812  df-un 2050  df-in 2051  df-ss 2053  df-sn 2412  df-pr 2413  df-tp 2415

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