Theorem tpss 1855

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

Hypotheses

Ref	Expression
tpss.1	⊢ A ∈ V
tpss.2	⊢ B ∈ V
tpss.3	⊢ C ∈ V

Assertion

Ref	Expression
tpss	⊢ ((A ∈ D ∧ B ∈ D ∧ C ∈ D) ↔ {A, B, C} ⊆ D)

Proof of Theorem tpss

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

Syntax hints:   → wi 2   ↔ wb 127   ∨ w3o 580   ∧ w3a 581   = wceq 1091   ∈ 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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