Theorem tpi2 2460

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 2455	. 2 |- B e. {A, B}
3		elun1 2200	. . 3 |- (B e. {A, B} -> B e. ({A, B} u. {C}))
4		df-tp 2419	. . 3 |- {A, B, C} = ({A, B} u. {C})
5	3, 4	syl6eleqr 1562	. 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 960  Vcvv 1814   u. cun 2048  {csn 2413  {cpr 2414  {ctp 2418

This theorem is referenced by:  tpss 2480  fr3nr 2932

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-v 1815  df-un 2053  df-in 2054  df-ss 2056  df-sn 2416  df-pr 2417  df-tp 2419

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