Theorem tpss 1855

Description: A triplet of elements of a class is a subset of the class.

Hypotheses

Ref	Expression
tpss.1	|- A e. V
tpss.2	|- B e. V
tpss.3	|- C e. V

Assertion

Ref	Expression
tpss	|- ((A e. D /\ B e. D /\ C e. D) <-> {A, B, C} (_ D)

Proof of Theorem tpss

Step	Hyp	Ref	Expression
1		3jao 632	. . . . 5 |- (((x = A -> x e. D) /\ (x = B -> x e. D) /\ (x = C -> x e. D)) -> ((x = A \/ x = B \/ x = C) -> x e. D))
2		eleq1a 1158	. . . . 5 |- (A e. D -> (x = A -> x e. D))
3		eleq1a 1158	. . . . 5 |- (B e. D -> (x = B -> x e. D))
4		eleq1a 1158	. . . . 5 |- (C e. D -> (x = C -> x e. D))
5	1, 2, 3, 4	syl3an 628	. . . 4 |- ((A e. D /\ B e. D /\ C e. D) -> ((x = A \/ x = B \/ x = C) -> x e. D))
6		visset 1350	. . . . 5 |- x e. V
7	6	eltp 1834	. . . 4 |- (x e. {A, B, C} <-> (x = A \/ x = B \/ x = C))
8	5, 7	syl5ib 181	. . 3 |- ((A e. D /\ B e. D /\ C e. D) -> (x e. {A, B, C} -> x e. D))
9	8	ssrdv 1509	. 2 |- ((A e. D /\ B e. D /\ C e. D) -> {A, B, C} (_ D)
10		tpss.1	. . . . 5 |- A e. V
11	10	tpi1 1843	. . . 4 |- A e. {A, B, C}
12		ssel 1502	. . . 4 |- ({A, B, C} (_ D -> (A e. {A, B, C} -> A e. D))
13	11, 12	mpi 44	. . 3 |- ({A, B, C} (_ D -> A e. D)
14		tpss.2	. . . . 5 |- B e. V
15	14	tpi2 1844	. . . 4 |- B e. {A, B, C}
16		ssel 1502	. . . 4 |- ({A, B, C} (_ D -> (B e. {A, B, C} -> B e. D))
17	15, 16	mpi 44	. . 3 |- ({A, B, C} (_ D -> B e. D)
18		tpss.3	. . . . 5 |- C e. V
19	18	tpi3 1845	. . . 4 |- C e. {A, B, C}
20		ssel 1502	. . . 4 |- ({A, B, C} (_ D -> (C e. {A, B, C} -> C e. D))
21	19, 20	mpi 44	. . 3 |- ({A, B, C} (_ D -> C e. D)
22	13, 17, 21	3jca 604	. 2 |- ({A, B, C} (_ D -> (A e. D /\ B e. D /\ C e. D))
23	9, 22	impbi 139	1 |- ((A e. D /\ B e. D /\ C e. D) <-> {A, B, C} (_ D)

Syntax hints:   -> wi 2   <-> wb 127   \/ w3o 580   /\ w3a 581   = wceq 1091   e. wcel 1092  Vcvv 1348   (_ wss 1487  {ctp 1813

This theorem is referenced by:  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-3or 582  df-3an 583  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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