Theorem tpss 3871

Description: A triplet of elements of a class is a subset of the class. (Contributed by NM, 9-Apr-1994.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)

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		unss 3437	. 2 ⊢ (({A, B} ⊆ D ∧ {C} ⊆ D) ↔ ({A, B} ∪ {C}) ⊆ D)
2		df-3an 936	. . 3 ⊢ ((A ∈ D ∧ B ∈ D ∧ C ∈ D) ↔ ((A ∈ D ∧ B ∈ D) ∧ C ∈ D))
3		tpss.1	. . . . 5 ⊢ A ∈ V
4		tpss.2	. . . . 5 ⊢ B ∈ V
5	3, 4	prss 3861	. . . 4 ⊢ ((A ∈ D ∧ B ∈ D) ↔ {A, B} ⊆ D)
6		tpss.3	. . . . 5 ⊢ C ∈ V
7	6	snss 3838	. . . 4 ⊢ (C ∈ D ↔ {C} ⊆ D)
8	5, 7	anbi12i 678	. . 3 ⊢ (((A ∈ D ∧ B ∈ D) ∧ C ∈ D) ↔ ({A, B} ⊆ D ∧ {C} ⊆ D))
9	2, 8	bitri 240	. 2 ⊢ ((A ∈ D ∧ B ∈ D ∧ C ∈ D) ↔ ({A, B} ⊆ D ∧ {C} ⊆ D))
10		df-tp 3743	. . 3 ⊢ {A, B, C} = ({A, B} ∪ {C})
11	10	sseq1i 3295	. 2 ⊢ ({A, B, C} ⊆ D ↔ ({A, B} ∪ {C}) ⊆ D)
12	1, 9, 11	3bitr4i 268	1 ⊢ ((A ∈ D ∧ B ∈ D ∧ C ∈ D) ↔ {A, B, C} ⊆ D)

Syntax hints:   ↔ wb 176   ∧ wa 358   ∧ w3a 934   ∈ wcel 1710  Vcvv 2859   ∪ cun 3207   ⊆ wss 3257  {csn 3737  {cpr 3738  {ctp 3739

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-ss 3259  df-sn 3741  df-pr 3742  df-tp 3743

This theorem is referenced by: (None)