Theorem tpss 4400

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 3820	. 2 ⊢ (({𝐴, 𝐵} ⊆ 𝐷 ∧ {𝐶} ⊆ 𝐷) ↔ ({𝐴, 𝐵} ∪ {𝐶}) ⊆ 𝐷)
2		df-3an 1056	. . 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 4383	. . . 4 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷) ↔ {𝐴, 𝐵} ⊆ 𝐷)
6		tpss.3	. . . . 5 ⊢ 𝐶 ∈ V
7	6	snss 4348	. . . 4 ⊢ (𝐶 ∈ 𝐷 ↔ {𝐶} ⊆ 𝐷)
8	5, 7	anbi12i 733	. . 3 ⊢ (((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷) ∧ 𝐶 ∈ 𝐷) ↔ ({𝐴, 𝐵} ⊆ 𝐷 ∧ {𝐶} ⊆ 𝐷))
9	2, 8	bitri 264	. 2 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷 ∧ 𝐶 ∈ 𝐷) ↔ ({𝐴, 𝐵} ⊆ 𝐷 ∧ {𝐶} ⊆ 𝐷))
10		df-tp 4215	. . 3 ⊢ {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶})
11	10	sseq1i 3662	. 2 ⊢ ({𝐴, 𝐵, 𝐶} ⊆ 𝐷 ↔ ({𝐴, 𝐵} ∪ {𝐶}) ⊆ 𝐷)
12	1, 9, 11	3bitr4i 292	1 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷 ∧ 𝐶 ∈ 𝐷) ↔ {𝐴, 𝐵, 𝐶} ⊆ 𝐷)

Syntax hints:   ↔ wb 196   ∧ wa 383   ∧ w3a 1054   ∈ wcel 2030  Vcvv 3231   ∪ cun 3605   ⊆ wss 3607  {csn 4210  {cpr 4212  {ctp 4214

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-v 3233  df-un 3612  df-in 3614  df-ss 3621  df-sn 4211  df-pr 4213  df-tp 4215

This theorem is referenced by:  1cubr  24614  konigsberglem4  27233  rabren3dioph  37696  fourierdlem102  40743  fourierdlem114  40755  nnsum4primesodd  42009  nnsum4primesoddALTV  42010