Theorem tpi3 1845

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

Hypothesis

Ref	Expression
tpi3.1	⊢ C ∈ V

Assertion

Ref	Expression
tpi3	⊢ C ∈ {A, B, C}

Proof of Theorem tpi3

Step	Hyp	Ref	Expression
1		tpi3.1	. . 3 ⊢ C ∈ V
2	1	snid 1830	. 2 ⊢ C ∈ {C}
3		elun2 1626	. . 3 ⊢ (C ∈ {C} → C ∈ ({A, B} ∪ {C}))
4		df-tp 1814	. . . 4 ⊢ {A, B, C} = ({A, B} ∪ {C})
5	4	eleq2i 1153	. . 3 ⊢ (C ∈ {A, B, C} ↔ C ∈ ({A, B} ∪ {C}))
6	3, 5	sylibr 175	. 2 ⊢ (C ∈ {C} → C ∈ {A, B, C})
7	2, 6	ax-mp 6	1 ⊢ C ∈ {A, B, C}

Syntax hints:   ∈ wcel 1092  Vcvv 1348   ∪ cun 1485  {csn 1808  {cpr 1809  {ctp 1813

This theorem is referenced by:  tpss 1855  fr3nr 2178

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-un 1490  df-in 1491  df-ss 1492  df-sn 1811  df-pr 1812  df-tp 1814

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